An Introduction toHybrid Dynamical Systems

Arjan van der SchaftDepartment of Systems, Signals and ControlFaculty of Mathematical SciencesUniversity of TwenteHans SchumacherCWI (Centre for Mathematics and Computer Science)AmsterdamandDepartment of Econometrics andCenter for Economic Research, Tilburg University

PrefaceThe term \hybrid system" has many meanings, one of which is: a dynamicalsystem whose evolution depends on a coupling between variables that takevalues in a continuum and variables that take values in a �nite or countable set.For a typical example of a hybrid system in this sense, consider a temperaturecontrol system consisting of a heater and a thermostat. Variables that would inall likelihood be included in a model of such a system are the room temperatureand the operating mode of the heater (on or off). It is natural to model the�rst variable as real-valued and the second as Boolean. Obviously, for thetemperature control system to be e�ective there needs to be a coupling betweenthe continuous and discrete variables, so that for instance the operating modewill be switched to on if the room temperature decreases below a certain value.Actually most of the dynamical systems that we have around us may rea-sonably be described in hybrid terms: cars, computers, airplanes, washingmachines|there is no lack of examples. Nevertheless, most of the literatureon dynamic modeling is concerned with systems that are either completelycontinuous or completely discrete. There are good reasons for choosing adescription in either the continuous or the discrete domain. Indeed, it is aplatitude that it is not necessary or even advisable to include all aspects of agiven physical system into a model that is intended to answer certain typesof questions. The engineering solution to a hybrid system problem thereforehas often been to look for a formulation that is primarily continuous or dis-crete, and to deal with aspects from the other domain, if necessary, in an adhoc manner. As a consequence, the �eld of hybrid system modeling has beendominated by patches and workarounds.Indications are, however, that the interaction between discrete and con-tinuous systems in today's technological problems has become so importantthat more systematic ways of dealing with hybrid systems are called for. Fora dramatic example, consider the loss of the Ariane 5 launcher that went intoself-destruction mode 37 seconds after lifto� on June 4, 1996. Investigatorshave put the blame for the costly failure on a software error. Nevertheless,the program that went astray was the same as the one that had worked per-fectly in Ariane 4; in fact, it was copied from Ariane 4 to Ariane 5 for exactlythat reason. What had changed was the continuous dynamical system aroundthe software, embodied in the physical structure of the new launcher whichv

vi Prefacehad been sized up considerably compared to its predecessor. Within the newphysical environment, the trusted code quickly led into a catastrophe.Although the increasing role of the computer in the control of physicalprocesses may be cited as one of the reasons for the increased interest in hybridsystems, there are also other sources of inspiration. In fact a number of recentdevelopments all revolve in some way around the combination of continuousand discrete aspects. The following is a sample of these developments, whichare connected in ways that are still largely unexplored:- computer science: veri�cation of correctness of programs interacting withcontinuous environments (embedded systems);- control theory: hierarchical control, interaction of data streams andphysical processes, stabilization of nonlinear systems by switching con-trol;- dynamical systems: discontinuous systems show new types of bifurca-tions and provide relatively tractable examples of chaos;- mathematical programming: optimization and equilibrium problemswith inequality constraints can fruitfully be placed within a regime-switching dynamic framework;- simulation languages: element libraries contain both continuous and dis-crete elements, so that the numerical simulation routines behind thelanguages must take both aspects into account.It is a major challenge to advance and systematize the knowledge about hybridsystems that comes from such a large variety of �elds, which nevertheless froma historical point of view have many concepts in common.Even though the overall area of hybrid systems has not yet crystallized,we believe that it is meaningful at this time to take stock of a number ofrelated developments in an introductory text, and to describe these from amore uni�ed point of view. We have not tried to be encyclopaedic, and inany case we consider the present text as an intermediate product. Our ownbackground is clearly re ected in the choice of the developments covered, andwithout doubt the reader will recognize a de�nite emphasis on aspects that areof interest from the point of view of continuous dynamics and mathematicalsystems theory. The title that we have chosen is intended to re ect this choice.We trust that others who are more quali�ed to do so will write books on hybridsystems emphasizing di�erent aspects.This text is an expanded and revised version of course notes that we havewritten for the course in Hybrid Systems that we taught in the spring of 1998as part of the national graduate course program of the Dutch Institute of Sys-tems and Control. We would like to thank the board of DISC for giving usthe opportunity to present this course, and we are grateful to the course par-ticipants for their comments. In some parts of the book we have relied heavilyon work that we have done jointly with Kanat C�aml�bel, Gerardo Escobar,

Preface viiMaurice Heemels, Jun-Ichi Imura, Yvonne Lootsma, Romeo Ortega, and SiepWeiland; it is a pleasure to acknowledge their contributions. The second au-thor would in particular like to thank Gjerrit Meinsma for his patient guidanceon the intricacies of LATEX. Finally, for their comments on preliminary ver-sions of this text and helpful remarks, we would like to thank Ren�e Boel, PeterBreedveld, Ed Brinksma, Bernard Brogliato, Domine Leenaerts, John Lygeros,Oded Maler, Sven-Erik Mattson, Gjerrit Meinsma, Manuel Monteiro Marques,Andrew Paice, Shankar Sastry, David Stewart, and Jan Willems. Of course,all remaining faults and fallacies are entirely our own.Enschede/Amsterdam, September 1999Arjan van der SchaftHans Schumacher

viii Preface

ContentsList of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiii1 Modeling of hybrid systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Towards a de�nition of hybrid systems . . . . . . . . . . . . . . . . . . . . . 31.2.1 Continuous and symbolic dynamics . . . . . . . . . . . . . . . . . . 31.2.2 Hybrid automaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Features of hybrid dynamics . . . . . . . . . . . . . . . . . . . . . . . . 91.2.4 Generalized hybrid automaton . . . . . . . . . . . . . . . . . . . . . . 121.2.5 Hybrid time evolutions and hybrid behavior . . . . . . . . . . 141.2.6 Event- ow formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.7 Simulation of hybrid systems . . . . . . . . . . . . . . . . . . . . . . . 241.2.8 Representations of hybrid systems . . . . . . . . . . . . . . . . . . . 301.3 Notes and References for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . 342 Examples of hybrid dynamical systems : : : : : : : : : : : : : : : : : : : : 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.1 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.2 Manual transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.3 Bouncing ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.4 Temperature control system . . . . . . . . . . . . . . . . . . . . . . . . 382.2.5 Water-level monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.6 Multiple collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.7 Variable-structure system . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2.8 Supervisor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.9 Two carts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.10 Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2.11 Systems with piecewise linear elements . . . . . . . . . . . . . . . 502.2.12 Railroad crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2.13 Power converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.14 Constrained pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2.15 Degenerate Van der Pol oscillator . . . . . . . . . . . . . . . . . . . 552.3 Notes and References for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 56ix

x Contents3 Variable-structure systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 573.1 Discontinuous dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Solution concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3 Reformulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4 Systems with many regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5 The Vidale-Wolfe advertizing model . . . . . . . . . . . . . . . . . . . . . . . 653.6 Notes and references for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 674 Complementarity systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 714.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1.1 Circuits with ideal diodes . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1.2 Mechanical systems with unilateral constraints . . . . . . . . 744.1.3 Optimal control with state constraints . . . . . . . . . . . . . . . 754.1.4 Variable-structure systems . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1.5 A class of piecewise linear systems . . . . . . . . . . . . . . . . . . . 794.1.6 Projected dynamical systems . . . . . . . . . . . . . . . . . . . . . . . 814.1.7 Di�usion with a free boundary . . . . . . . . . . . . . . . . . . . . . . 844.1.8 Max-plus systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . 904.3 The mode selection problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.4 Linear complementarity systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.4.1 Speci�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.4.2 A distributional interpretation . . . . . . . . . . . . . . . . . . . . . . 1004.4.3 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5 Mechanical complementarity systems . . . . . . . . . . . . . . . . . . . . . . . 1044.6 Relay systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.7 Notes and references for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 1105 Analysis of hybrid systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1115.1 Correctness and reachability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1.1 Formal veri�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1.2 An audio protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1.3 Algorithms for veri�cation . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.2.1 Lyapunov functions and Poincar�e mappings . . . . . . . . . . . 1185.2.2 Time-controlled switching . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.2.3 State-controlled switching . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Chaotic phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.4 Notes and references for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 1326 Hybrid control design : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1336.1 Safety and guarantee properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.1.1 Safety and controlled invariance . . . . . . . . . . . . . . . . . . . . . 1356.1.2 Safety and dynamic game theory . . . . . . . . . . . . . . . . . . . . 1386.2 Switching control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.2.1 Switching logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Contents xi6.2.2 PWM control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.2.3 Sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.2.4 Quadratic stabilization by switching control . . . . . . . . . . 1466.3 Hybrid feedback stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.3.1 Energy decrease by hybrid feedback. . . . . . . . . . . . . . . . . . 1486.3.2 Stabilization of nonholonomic systems . . . . . . . . . . . . . . . 1516.3.3 Set-point regulation of mechanical systems by energyinjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.4 Notes and References for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 158Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 159Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 171

xii Contents

List of Figures1.1 Finite automaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Hybrid automaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Solutions of Filippov's example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.5 Partitioning of the plane induced by implicit scheme . . . . . . . . . . . . . 312.1 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Control system with hysteresis as a hybrid automaton . . . . . . . . . . . . 362.3 Temperature-thermostat system as a hybrid automaton . . . . . . . . . . 392.4 Water-level monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Collision to an elastic wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Supervisor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.7 Two carts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.8 Coulomb friction characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.9 Boost circuit with clamping diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.10 Ideal diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.11 Pendulum constrained by a pin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1 Trajectories for Vidale-Wolfe example. . . . . . . . . . . . . . . . . . . . . . . . . . 684.1 Values of an American put option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2 Two carts with stop and hook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.3 Tangent plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1 Reachable set and avoidance set for audio protocol . . . . . . . . . . . . . . 1165.2 Trajectory of a switched system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3 Trajectories of two linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.1 Switching control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.2 Switching control with shared controller state . . . . . . . . . . . . . . . . . . . 1416.3 Hybrid feedback for the harmonic oscillator . . . . . . . . . . . . . . . . . . . . . 1496.4 An alternative hybrid feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.5 A typical partition of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155xiii

xiv List of Figures

Chapter 1Modeling of hybrid systems1.1 IntroductionThe aim of this chapter is to make more precise what we want to understand bya \hybrid system". This will be done in a somewhat tentative manner, withoutactually ending up with a single �nal de�nition of a hybrid system. Partly, thisis due to the fact that the area of hybrid systems is still in its infancy and thata general theory of hybrid systems seems premature. More inherently, hybridsystems is such a wide notion that sticking to a single de�nition shall be toorestrictive (at least at this moment) for our purposes. Moreover, the choice ofhybrid models crucially depends on their purpose, e.g. for theoretical purposesor for speci�cation purposes, or as a simulation language. Nevertheless, wehope to make reasonably clear what should be the main ingredients in anyde�nition of a hybrid system, by proposing and discussing in Section 1.2 anumber of de�nitions of hybrid systems. Subsequently, Chapter 2 will presenta series of examples of hybrid systems illustrating the main ingredients of thesede�nitions in more detail.Generally speaking, hybrid systems are mixtures of real-time (continuous)dynamics and discrete events. These continuous and discrete dynamics notonly coexist, but interact and changes occur both in response to discrete,instantaneous, events and in response to dynamics as described by di�erentialor di�erence equations in time.One of the main di�culties in the discussion of hybrid systems is thatthe term \hybrid" is not restrictive|the interpretation of the term couldbe stretched to include virtually any dynamical system we can think of. Areasonably general de�nition of hybrid systems can therefore only serve as aframework, to indicate the main issues and to �x the terminology. Withinsuch a general framework one necessarily has to restrict to special subclassesof hybrid systems in order to derive useful generally valid propositions.Another di�culty in discussing hybrid systems is that various scienti�ccommunities with their own approaches have contributed (and are still con-tributing) to the area. At least the following three communities can be distin-guished.First there is the computer science community that looks at a hybrid sys-tem primarily as a discrete (computer) program interacting with an analog1

2 Chapter 1. Modeling of hybrid systemsenvironment. (In this context also the terminology embedded systems is beingused.) A leading objective is to extend standard program analysis techniquesto systems which incorporate some kind of continuous dynamics. The empha-sis is often on the discrete event dynamics, whereas the continuous dynamicsis frequently of a relatively simple form. One of the key issues is veri�cation.Another community involved in the study of hybrid systems is the model-ing and simulation community. Physical systems can often operate in di�erentmodes, and the transition from one mode to another sometimes can be idealizedas an instantaneous, discrete, transition. Examples include electrical circuitswith switching devices such as (ideal) diodes and transistors, and mechanicalsystems subject to inequality constraints as encountered e.g. in robotics. Sincethe time scale of the transition from one mode to another is often much fasterthan the time scale of the dynamics of the individual modes, it may be advan-tageous to model the transitions as being instantaneous. The time instant atwhich the transition takes place is called an event time. Basic issues then con-cern the well-posedness of the resulting hybrid system, e.g. the existence anduniqueness of solutions, and the ability to e�ciently simulate the multi-modalphysical system.Yet another community contributing to the area of hybrid systems is thesystems and control community. Within this community additional motiva-tion for the study of hybrid systems is actually provided from di�erent angles.One can think of hierarchical systems with a discrete decision layer and a con-tinuous implementation layer (e.g. supervisory control or multi-agent control).Also switching control schemes and relay control immediately lead to hybridsystems. For nonlinear control systems it is known that in some importantcases there does not exist a continuous stabilizing state feedback, but that nev-ertheless the system can be stabilized by a switching control. Finally, discreteevent systems theory can be seen as a special case of hybrid systems theory. Inmany areas of control, e.g. in power converters and in motion control, controlstrategies are inherently hybrid in nature.From a general system-theoretic point of view one can look at hybrid sys-tems as systems having two di�erent types of ports through which they interactwith their environment. One type of ports consists of the communication ports.The variables associated with these ports are symbolic in nature, and represent\data ow". The strings of symbols at these communication ports in generalare not directly related with real (physical) time; there is only a sequentialordering. The second type of ports consists of the physical ports, where theterm \physical" interpreted in a broad sense; perhaps \analog" would be amore appropriate terminology. The variables at these ports are usually con-tinuous variables, and are related to physical measurement. Also, the ow ofthese variables is directly related to physical time. In principle the signals atthe physical ports may be discrete-time signals (or sampled-data signals), butin most cases they will ultimately be continuous-time signals.Thus a hybrid system can be regarded as a combination of discrete or sym-bolic dynamics and continuous dynamics. The main problem in the de�nition

1.2. Towards a de�nition of hybrid systems 3and representation of a hybrid system is precisely to specify the interactionbetween this symbolic and continuous dynamics.A key issue in the formulation of hybrid systems is the often required mod-ularity of the hybrid system description. Indeed, because we are inherentlydealing with the modeling of complex systems, it is very important to model acomplex hybrid system as the interconnection of simpler (hybrid) subsystems.This implies that the hybrid models that we are going to discuss are prefer-ably of a form that admits easy interconnection and composition. Besidesthis notion of compositionality other important (related) notions are those of\reusability" and \hierarchy". These terms arise in the context of \object-oriented modeling".1.2 Towards a de�nition of hybrid systemsIn our opinion, from a conceptual point of view the most basic de�nition ofa hybrid system is to immediately specify its behavior, that is, the set of allpossible trajectories of the continuous and discrete variables associated withthe system. On the other hand, such a behavioral de�nition tends to be verygeneral and far from an operational speci�cation of hybrid systems. Instead weshall start with a reasonably generally accepted \working de�nition" of hybridsystems, which already has proved its usefulness. This de�nition, called thehybrid automaton model, will already provide the framework and terminologyto discuss a range of typical features of hybrid systems. At the end of thischapter we are then prepared to return to the issues that enter into a behavioralde�nition of hybrid systems, and to discuss an alternative way of modelinghybrid systems by means of equations.1.2.1 Continuous and symbolic dynamicsIn order to motivate the hybrid automaton de�nition, we recall the\paradigms" of continuous and symbolic dynamics; namely, state space mod-els described by di�erential equations for continuous dynamics, and �nite au-tomata for symbolic dynamics. Indeed, the de�nition of a hybrid automatonbasically combines these two paradigms. Note that both in the continuousdomain and in the discrete domain one can think of more general settings;for instance partial di�erential equations and stochastic di�erential equationson the continuous side, pushdown automata and Turing machines on the dis-crete side. The framework we shall discuss however is already suitable for themodeling of many applications, as will be shown in the next chapter.De�nition 1.2.1 (Continuous-time state-space models).A continuous-time state-space system is described by a set of state variablesx taking values in Rn (or, more generally, in an n-dimensional state spacemanifold X), and a set of external variables w taking values in Rq , related by

4 Chapter 1. Modeling of hybrid systemsa mixed set of di�erential and algebraic equations of the formF (x; _x;w) = 0: (1.1)Here _x denotes the derivative of x with respect to time. Solutions of (1.1) areall (su�ciently smooth) time functions x(t) and w(t) satisfyingF (x(t); _x(t); w(t)) = 0for (almost) all times t 2 R (the continuous-time axis).Of course, the above de�nition encompasses the more common de�nitionof a continuous-time input-state-output system_x = f(x; u)y = h(x; u) (1.2)where we have split the vector of external variables w into a subvector u takingvalues in Rm and a subvector y taking values in Rp (with m+ p = q), calledrespectively the vector of input variables and the vector of output variables.The only algebraic equations in (1.2) are those relating the output variables yto x and u, while generally in (1.1) there are additional algebraic constraintson the state space variables x.One of the main advantages of general continuous-time state space systems(1.1) over continuous-time input-state-output systems (1.2) is the fact that the�rst class is closed under interconnection, while the second class in general isnot. In fact, modeling approaches that are based on modularity (viewing thesystem as the interconnection of smaller subsystems) almost invariably leadto a mixed set of di�erential and algebraic equations. Of course, in a numberof cases it may be relatively easy to eliminate the algebraic equations in thestate space variables, in which case (if we can also easily split w into u and y)we can convert (1.1) into (1.2).We note that De�nition 1.2.1 does not yet completely specify thecontinuous-time system, since (on purpose) we have been rather vague aboutthe precise solution concept of the di�erential-algebraic equations (1.1). Forexample, a reasonable choice (but not the only possible one!) is to requirew(t) to be piecewise continuous (allowing for discontinuities in the \inputs")and x(t) to be continuous and piecewise di�erentiable, with (1.1) being sat-is�ed for almost all t (except for the points of discontinuity of w(t) and non-di�erentiability of x(t)).Next we give the standard de�nition of a �nite automaton (or �nite statemachine, or labeled transition system).De�nition 1.2.2 (Finite automaton). A �nite automaton is described bya triple (L;A;E). Here L is a �nite set called the state space, A is a �nite setcalled the alphabet whose elements are called symbols. E is the transition rule;it is a subset of L�A�L and its elements are called edges (or transitions, orevents). A sequence (l0; a0; l1; a1; : : : ; ln�1; an�1; ln) with (li; ai; li+1) 2 E fori = 1; 2; : : : ; n� 1 is called a trajectory or path.

1.2. Towards a de�nition of hybrid systems 5b

l1a b c

cad

l2 l4l3Figure 1.1: Finite automatonThe usual way of depicting an automaton is by a graph with vertices givenby the elements of L, and edges given by the elements of E, see Figure 1.1.Then A can be seen as a set of labels labeling the edges. Sometimes these arecalled synchronization labels, since interconnection with other automata takesplace via these (shared) symbols. One can also specialize De�nition 1.2.2 toinput-output automata by associating with every edge two symbols, namelyan input symbol i and an output symbol o, and by requiring that for everyinput symbol there is only one edge originating from the given state withthis input symbol. (Sometimes such automata are called deterministic input-output automata.) Deterministic input-output automata can be representedby equations of the following form:l] = �(l; i)o = �(l; i) (1.3)where l] denotes the new value of the discrete state after the event takes place,resulting from the old discrete state value l and the input i.Often the de�nition of a �nite automaton also includes the explicit speci-�cation of a subset I � L of initial states and a subset F � L of �nal states.A path (l0; a0; l1; a1; : : : ; ln�1; an�1; ln) is then called a successful path if inaddition l0 2 I and ln 2 F .In contrast with the continuous-time systems de�ned in De�nition 1.2.1 thesolution concept (or semantics) of a �nite automaton (with or without initialand �nal states) is completely speci�ed: the behavior of the �nite automatonconsists of all (successful) paths. In theoretical computer science parlance thede�nition of a �nite automaton is said to entail an \operational semantics",completely specifying the formal language generated by the �nite automaton.

6 Chapter 1. Modeling of hybrid systemsNote that the de�nition of a �nite automaton is conceptually not very dif-ferent from the de�nition of a continuous-time state space system. Indeedwe may relate the state space L with the state space X , the symbol alpha-bet A with the space W (where the external variables take their values), andthe transition rule E with the set of di�erential-algebraic equations given by(1.1). Furthermore the paths of the �nite automaton correspond to the so-lutions of the set of di�erential-algebraic equations. The analogy betweencontinuous-time input-state-output systems (1.2) and input-output automata(1.3) is obvious, with the di�erentiation operator ddt replaced by the \nextstate" operator ].A (minor) di�erence is that in �nite automata one usually considers (as inDe�nition 1.2.2) paths of �nite length, while for continuous-time state spacesystems the emphasis is on solutions over the whole time axis R. This could beremedied by adding to the �nite automaton a source state and a sink state anda blank label, and by considering solutions de�ned over the whole time axis Zwhich \start" at minus in�nity in the source state and \end" at plus in�nityin the sink state, while producing the blank symbol when remaining in thesource or sink state. Also the set I of initial states and the set F of �nal statesin some de�nitions of a �nite automaton do not have a direct analogon in thede�nition of a continuous-time state space system. In some sense, however,they could be viewed as a formulation of performance speci�cations of theautomaton.Summarizing, the basic di�erences between De�nition 1.2.1 and De�nition1.2.2 are the following.- The spaces L and A are �nite sets instead of continuous spaces such asX and W . (In some extensions one allows for countable sets L and A.)- The time axis in De�nition 1.2.1 is R, while the time axis in De�nition1.2.2 is Z. Here, to be precise, Z is understood without any structure ofaddition (only sequential ordering).- In the �nite automaton model the set of possible transitions (events) isspeci�ed explicitly, while the evolution of a continuous-time state-spacesystem is only implicitly given by the set of di�erential and algebraicequations (one still needs to solve these equations).1.2.2 Hybrid automatonCombining De�nitions 1.2.1 and 1.2.2 leads to the following type of de�nitionof a hybrid system.De�nition 1.2.3 (Hybrid automaton). A hybrid automaton is describedby a septuple (L;X;A;W;E; Inv;Act) where the symbols have the followingmeanings.- L is a �nite set, called the set of discrete states or locations . They arethe vertices of a graph.

1.2. Towards a de�nition of hybrid systems 7- X is the continuous state space of the hybrid automaton in which thecontinuous state variables x take their values. For our purposes X � Rnor X is an n-dimensional manifold.- A is a �nite set of symbols which serve to label the edges.- W = Rq is the continuous communication space in which the continuousexternal variables w take their values.- E is a �nite set of edges called transitions (or events). Every edge isde�ned by a �ve-tuple (l; a;Guardll0 ; Jumpll0 ; l0), where l; l0 2 L, a 2 A,Guardll0 is a subset of X and Jumpll0 is a relation de�ned by a subsetof X �X . The transition from the discrete state l to l0 is enabled whenthe continuous state x is in Guardll0 , while during the transition thecontinuous state x jumps to a value x0 given by the relation (x; x0) 2Jumpll0 .- Inv is a mapping from the locations L to the set of subsets of X , thatis Inv(l) � X for all l 2 L. Whenever the system is at location l, thecontinuous state x must satisfy x 2 Inv(l). The subset Inv(l) for l 2 Lis called the location invariant of location l.- Act is a mapping that assigns to each location l 2 L a set of di�erential-algebraic equations Fl, relating the continuous state variables x withtheir time-derivatives _x and the continuous external variables w:Fl(x; _x;w) = 0: (1.4)The solutions of these di�erential-algebraic equations are called the ac-tivities of the location.Clearly, the above de�nition draws strongly upon De�nition 1.2.2, the dis-crete state space L now being called the space of locations. (Note that theset of edges E in De�nition 1.2.3 also de�nes a subset of L � A � L.) Infact, De�nition 1.2.3 extends De�nition 1.2.2 by associating with every vertex(location) a continuous dynamics whose solutions are the activities, and byassociating with every transition l! l0 also a possible jump in the continuousstate.Note that the state of a hybrid automaton consists of a discrete part l 2 Land a continuous part in X . Furthermore, the external variables consist of adiscrete part taking their values a in A and a continuous part w taking theirvalues in Rq . Also, the dynamics consists of discrete transitions (from onelocation to another), together with a continuous part evolving in the locationinvariant.It should be remarked that the above de�nition of a hybrid automaton hasthe same ambiguity as the de�nition of a continuous-time state-space system,since it still has to be complemented by a precise speci�cation of the solutions(activities) of the di�erential-algebraic equations associated with every loca-tion. In fact, in the original de�nitions of a hybrid automaton (see e.g. [1]) the

8 Chapter 1. Modeling of hybrid systemsactivities of every location are assumed to be explicitly given rather than gen-erated implicitly as the solutions to the di�erential-algebraic equations. Onthe other hand, somebody acquainted with di�erential equations would not�nd it convenient in general to have to specify continuous dynamics immedi-ately by time functions from R+ to X . Indeed, continuous time dynamics isalmost always described by sets of di�erential or di�erential-algebraic equa-tions, and only in exceptional cases (such as linear dynamical systems) one canobtain explicit solutions. The description of a hybrid automaton is illustratedin Figure 1.2.

x(t) 2 Inv(`2)x(t) 2 Inv(`1)

F`3(x; _x;w) = 0x(t) 2 Inv(`3)`4F`4(x; _x;w) = 0x(t) 2 Inv(`4)F`2(x; _x; w) = 0F`1(x; _x;w) = 0a Guard ac

`1

b GuardJump Jump

Jumpx0 := 0Guardx(t) � �Guardc Jump `2

`3GuardJumpGuard

GuardJumpJump db

Figure 1.2: Hybrid automatonA reasonable de�nition of the trajectories (or solutions, or in computerscience terminology, the runs or executions) of a hybrid automaton can beformulated as follows. A continuous trajectory (l; �; x; w) associated with alocation l consists of a nonnegative time � (the duration of the continuoustrajectory), a piecewise continuous function w : [0; �]! W , and a continuousand piecewise di�erentiable function x : [0; �]! X such that

1.2. Towards a de�nition of hybrid systems 9- x(t) 2 Inv(l) for all t 2 (0; �),- Fl(x(t); _x(t); w(t)) = 0 for all t 2 (0; �) except for points of discontinuityof w.A trajectory of the hybrid automaton is an (in�nite) sequence of continuoustrajectories(l0; �0; x0; w0) a0! (l1; �1; x1; w1) a1! (l2; �2; x2; w2) a2! : : :such that at the event timest0 = �0; t1 = �0 + �1; t2 = �0 + �1 + �2; : : :the following inclusions hold for the discrete transitions:xj(tj) 2 Guardlj lj+1(xj(tj); xj+1(tj)) 2 Jumplj lj+1 for all j = 0; 1; 2; : : : :Furthermore, to the j-th arrow ! in the above sequence (with j starting at0) one associates a symbol (label) aj , representing the value of the discrete\signal" at the j-th discrete transition.1.2.3 Features of hybrid dynamicsNote that the trajectories of a hybrid automaton exhibit the following features.Starting at a given location the continuous part of the state evolves accordingto the continuous dynamics associated with this location, provided it remainsin the location invariant. Then, at some time instant in R, called an eventtime, an event occurs and the discrete part of the state (the location) switchesto another location. This is an instantaneous transition which is guarded, thatis, a necessary condition for this transition to take place is that the guard ofthis transition is satis�ed. Moreover in general this transition will also involvea jump in the continuous part of the state. Then, after the instantaneoustransition has taken place, the continuous part of the state, starting fromthis new continuous state, will in principle evolve according to the continuousdynamics of the new location. Thus there are two phenomena associated withevery event, a switch and a jump, describing the instantaneous transition of,respectively, the discrete and the continuous part of the state at such an eventtime.A basic issue in the speci�cation of a hybrid system is the speci�cationof the events and event times. First, the events may be externally inducedvia the labels (symbols) a 2 A; this leads to controlled switchings and jumps.Secondly, the events may be internally induced ; this leads to what is called au-tonomous switchings and jumps. The occurrence of internally induced eventsis determined by the guards and the location invariants. Whenever violationof the location invariants becomes imminent, the hybrid automation has to

10 Chapter 1. Modeling of hybrid systemsswitch to a new location, with possibly a reset of the continuous state. Atsuch an event time the guards will determine to which locations the transitionis possible. (There may be more than one; furthermore, it may be possible toswitch to the same location.)If violation of the location invariants is not imminent then still discretetransitions may take place if the corresponding guards are satis�ed. That is,if at a certain time instant the guard of a discrete transition is satis�ed thenthis may cause an event to take place. As a result one may obtain a largeclass of trajectories of the hybrid automaton, and a tighter speci�cation of thebehavior of the hybrid automaton will critically depend on a more restrictivede�nition of the guards. Intuitively the location invariants provide enforcingconditions whereas the guards provide enabling conditions.Many issues naturally come up in connection with the analysis of the tra-jectories of a hybrid automaton. We list a number of these.- It could happen that, after some time t, the system ends up in a state(l; x(t)) from which there is no continuation, that is, there is no pos-sible continuous trajectory from x(t) and no possible transition to an-other location. In the computer science literature this is usually called\deadlock". Usually this will be considered an undesirable phenomenon,because it means that the system is \stuck".- The set of durations �i may get smaller and smaller for increasing i evento such an extent that Pi=1i=0 �i is �nite, say � . This means that � is anaccumulation point of event times. In the computer science literaturethis is called Zeno behavior (referring to Zeno's paradox of Achilles andthe turtle). This need not always be an undesirable behavior of thehybrid system. In fact, as long as the continuous and discrete parts ofthe state will converge to a unique value at the accumulation point � ,then we can re-initialize the hybrid system at � using these limits, andlet the system run as before starting from the initial time � . In fact, inSubsection 2.2.3 (the bouncing ball) we will encounter a situation likethis.- In principle the durations of the continuous trajectories are allowed to bezero; in this way one may cover the occurrence of multiple events . In thiscase the underlying time axis of the hybrid trajectory has a structurewhich is more complicated than R containing a set of event times: acertain time instant t 2 R may correspond to a sequence of sequentiallyordered transitions, all happening at the same time instant t which isthen called a multiple event time. We refer e.g. to Subsection 2.2.6 wherewe also provide an example of an event with multiplicity 1.- It may happen that the hybrid system gets stuck at such a multipleevent time by switching inde�nitely between di�erent locations (and notproceeding in time). This is sometimes called livelock . Such a situationoccurs if a location invariant tends to be violated and a guarded transi-tion takes place to a new location in such a way that the new continuous

1.2. Towards a de�nition of hybrid systems 11state does not satisfy the location invariants of the new location, whilethe guard for the transition to the old location is satis�ed. In some casesthis problem can be resolved by creating a new location, with a continu-ous dynamics that \averages" the continuous dynamics of the locationsbetween which the in�nite switching occurs. Filippov's notions of so-lutions of discontinuous vector �elds can be interpreted in this way, cf.Subsection 2.2.7 and Chapter 3.- In general the set of trajectories (runs) of the hybrid automaton maybe very large, especially if the guards are not very strict. For certainpurposes (e.g. veri�cation) this may not be a problem, but in other casesone may wish the hybrid system to be deterministic or well-posed , inthe sense of having unique solutions for given discrete and continuous\inputs" (assuming that we have split the vector w of continuous externalvariables into a vector of continuous inputs and outputs, and that everylabel a actually consists of an input label and an output label). Especiallyfor simulation purposes this may be very desirable, and in this contextone would often dismiss a hybrid model as inappropriate if it does nothave this well-posedness property. On the other hand, nondeterministicdiscrete-event systems are very common in computer science. A simpleexample of a hybrid system not having unique solutions, at least from acertain point of view, is provided by the integrator dynamics_y = uin conjunction with the relay elementu = +1; y > 0u = �1; y < 0�1 � u � 1; y = 0:The resulting hybrid system (without external inputs) has three loca-tions corresponding to the three segments of the ideal relay element. Itis directly seen that for initial condition y(0) = 0 the system can evolvein either of these three locations, yielding the three solutions (i) y(t) = t,u(t) = 1, (ii) y(t) = �t, u(t) = �1, and (iii) y(t) = 0, u(t) = 0. Hence, ifwe specify the initial condition of this hybrid system only by its continu-ous state, which is not unreasonable in physical systems, then there arethree solutions starting from the zero initial condition. More involvedexamples of the same type will be given in Chapter 4. A \physical"example of the same type exhibiting non-uniqueness of solutions is theclassical example, due to Painlev�e, of a stick that slides with one endon a table and that is subject to Coulomb friction acting at the contactpoint; see e.g. [31, p. 154].- The continuous dynamics associated to a location may have \�nite escapetime", in the sense that the solution of the di�erential equations, or of the

12 Chapter 1. Modeling of hybrid systemsdi�erential-algebraic equations, for some initial state goes to \in�nity"(or to the boundary of the state space X) in �nite time. This is awell-known phenomenon in nonlinear di�erential equations. A simpleexample is provided by the di�erential equation_x(t) = 1 + x2(t); x(0) = 0;a solution of which is x(t) = tan t. (The framework of hybrid systemsallows one, in principle, to model this as an event, by letting the \explo-sion time" (�2 in the above example) be an event time, where the systemmay switch to a new location and jump to a new continuous state.)- The solution concept of the continuous-time dynamics associated to alocation may itself be problematic, especially because of the possiblepresence of algebraic constraints. In particular, in some situations onemay want to associate jump behavior with these continuous-time dynam-ics (see e.g. Subsection 2.2.15). Within the hybrid framework this canbe incorporated as internally induced events, where the system switchesto the same location but is subject to a reset of the continuous state.Remark 1.2.4. Of course, the de�nition of a hybrid automaton can still begeneralized in a number of directions. A particularly interesting extension is toconsider stochastic hybrid systems, such as the systems described by piecewise-deterministic Markov processes, see e.g. [40]. In this notion the event times aredetermined by the system reaching certain boundaries in the continuous statespace (similar to the notion of location invariants), and/or by an underlyingprobability distribution. Furthermore, also the resulting discrete transitionstogether with their jump relations are assumed to be governed by a probabilitydistribution.1.2.4 Generalized hybrid automatonIn De�nition 1.2.3 of a hybrid automaton there is still an apparent asymmetrybetween the continuous and the symbolic (discrete) part of the dynamics.Furthermore, the location invariants and the guards play strongly related rolesin the speci�cation of the discrete transitions. The following generalization ofDe�nition 1.2.3 takes the location invariants and the set of edges E together,and symmetrizes the de�nition of a hybrid automaton. (The input-outputversion of this de�nition is due to [98].)De�nition 1.2.5 (Generalized hybrid automaton). A generalized hy-brid automaton is described by a sixtuple (L;X;A;W;R;Act) whereL;X;A;W and Act are as in De�nition 1.2.3, and R is a subset of (L�X)�(A�W )� (L�X).A continuous trajectory (l; a; �; x; w) associated with a location l and adiscrete external symbol a consists of a nonnegative time � (the duration ofthe continuous trajectory), a piecewise continuous function w : [0; �] ! W ,and a continuous and piecewise di�erentiable function x : [0; �]! X such that

1.2. Towards a de�nition of hybrid systems 13- (l; x(t); a; w(t); l; x(t)) 2 R for all t 2 (0; �),- Fl(x(t); _x(t); w(t)) = 0 for almost all t 2 (0; �) (exceptions include thepoints of discontinuity of w).A trajectory of the generalized hybrid automaton is an (in�nite) sequence(l0; a0; �0; x0; w0)! (l1; a1; �1; x1; w1)! (l2; a2; �2; x2; w2)! : : :such that at the event timest0 = �0; t1 = �0 + �1; t2 = �0 + �1 + �2; : : :the following relations hold:(lj ; xj(tj); aj ; wj(tj); lj+1; xj+1(tj)) 2 R; for all j = 0; 1; 2; : : :The subset R incorporates the notions of the location invariants, guards, andjumps of De�nition 1.2.3 in the following way. To each location l we associatethe location invariantInv(l) = f(x; a; w) 2 X �A�W j (l; x; a; w; l; x) 2 Rg:(With an abuse of notation, x and w denote here elements of X , respectivelyW , instead of variables taking their values in these spaces.) Furthermore,given two locations l; l0 we obtain the following guard for the transition froml to l0:Guardll0 = f(x; a; w) 2 X �A�W j 9x0 2 X; (l; x; a; w; l0; x0) 2 Rgwith the interpretation that the transition from l to l0 can take place if andonly if (x; a; w) 2 Guardll0 . Finally, the associated jump relation is given byJumpll0(x; a; w) = fx0 2 X j (l; x; a; w; l0; x0) 2 Rg:Note that the resulting location invariants, guards as well as jump relationsare in principle of a more general type than in De�nition 1.2.3, since they allmay depend on the continuous and discrete external variables. (In Subsection2.2.13 of Chapter 2 we shall see that this is actually an appropriate generaliza-tion.) Conversely, it can be readily seen that any set E of edges and locationinvariants as in De�nition 1.2.3 can be recovered from a suitably chosen setR as in De�nition 1.2.5. Therefore, De�nition 1.2.5 does indeed generalizeDe�nition 1.2.3.A further generalization of the de�nition of a hybrid automaton would beto allow di�erent continuous state spaces associated with di�erent locations.(This is now to some extent captured in the location invariants or the subsetR.) In fact, some of the examples that we will present in Chapter 2 do motivatesuch a generalization.

14 Chapter 1. Modeling of hybrid systemsThe de�nition of a (generalized) hybrid automaton admits compositionalityin the following sense (cf. [1]). For simplicity we shall only give the de�nitionsfor the generalized hybrid automaton model (De�nition 1.2.5). Consider twogeneralized hybrid automata �i = (Li; Xi; Ai;Wi; Ri; Acti), i = 1; 2, and sup-pose thatW1 =W2; A1 = A2(This is the case of shared external variables.) We de�ne the synchronous par-allel composition or interconnection � = �1 k �2 of the two generalized hybridautomata �i; i = 1; 2, as the generalized hybrid automaton (L;X;A;W;R;Act)given byL = L1 � L2X = X1 �X2A = A1 = A2W =W1 =W2R = f((l1; l2); (x1; x2); a; w; (l01; l02); (x01; x02)) 2 (L�X)�(A�W )� (L�X) j (li; xi; a; w; l0i; x0i) 2 Ri; i = 1; 2gAct = Act1 �Act2: (1.5)This corresponds to the full synchronous parallel composition or interconnec-tion of the two generalized hybrid automata. We may also consider partialsynchronous parallel compositions by synchronizing only a part of the discreteexternal variables a1 and in a2 in A1, respectively A2, and by considering moregeneral relations between the continuous external variables w1 and w2. Fur-thermore, we may also de�ne the interleaving parallel composition � = �1kj�2by takingA = A1�A2 andW =W1�W2 in (1.5), and by de�ning R = R1�R2.1.2.5 Hybrid time evolutions and hybrid behaviorA conceptual problem in De�nitions 1.2.3 and 1.2.5 is the formalization ofthe notion of the time evolution corresponding to a hybrid trajectory, and inparticular the embedding of the event times in the continuous time axis R.In Def. 1.2.5 time is broken up into a sequence of closed intervals, which mayreduce to single points. E�ectively a counting label is added to time pointswhich indicates to which element of the sequence the point belongs; this isnecessary to make it possible that state variables have di�erent values at the\same" event time. A similar labeling procedure has been used for instancein [162], where the authors speak of a \time space". These notions of timeare not able to cover situations in which the event times of a trajectory havean accumulation point but still the trajectory does progress in time after this

1.2. Towards a de�nition of hybrid systems 15accumulation point. Examples of such a situation are provided in Subsections2.2.3 and 2.2.6. Therefore we propose here a somewhat more general notion.Let R be the continuous time axis. The time evolution corresponding to ahybrid trajectory will be speci�ed by a set E of time events . A time event inE consists of an event time t 2 R together with a multiplicity m(t), which isan element of N [ 1, where N is the set of natural numbers 1; 2; 3; : : : . Thetime event will be denoted by the sequence(t0]; t1]; t2]; : : : ; tm(t)])specifying the sequentially ordered \discrete transition times" at the samecontinuous time instant t 2 R (the event time). For simplicity of notation wewill sometimes write t]; t]]; t]]]; : : : for t1]; t2]; t3]; : : : .A time event with multiplicity equal to 1 is just given by a pair(t0]; t])with the interpretation of denoting the time instants \just before" and \justafter" the event has taken place (a more formal description will be given ina moment). If the multiplicity of the time event is larger than 1 (a multipletime event) then there are some \intermediate time instants" (\all at the sameevent time t") ordering the sequence of discrete transitions taking place at t.The \embedding" of the discrete dynamics into the continuous dynamicswill be performed by the compatibility conditionsx(t+) = x(tm(t)])x(t�) = x(t0]);where, in a usual notation,x(t+) := lim�#t x(�)x(t�) := lim�"t x(�):Note that events are allowed to have multiplicity equal to 1, in which casethere are an in�nite number of discrete transitions taking place at the samecontinuous time instant t. It seems natural to require that in such a situationthe sequence of locations l1; l2; l3; : : : at this time t converges to a single lo-cation, and that limk!1 x(tk]) exists (or, perhaps, has only a �nite numberof accumulation points). An example of such a situation will be provided inSubsection 2.2.6.The set of event times corresponding to a set of time events E will bedenoted by ET � R. Hence the set E of time events can be written asE = [t2ET f(t0]; t1]; : : : ; tm(t)])g:For the moment we will allow ET to be an arbitrary subset of R, but lateron (when dealing with the solution concept of a hybrid system) we will putrestrictions on ET .

16 Chapter 1. Modeling of hybrid systemsThe time evolution �E corresponding to a time event set E is de�ned to bethe set�E = (R n ET ) [ E :The union of all time evolutions �E for all time event sets E will be denotedby T . The notion of a time evolution is illustrated in Figure 1.3.t]3t0]1 t0]2 t0]3t2t1

t]]]2t3t]]2t]2t]]1t]1

Figure 1.3: Time evolutionWe now formalize the notion of a trajectory of a generalized hybrid au-tomaton using the concept of time evolutions. For clarity of notation, we willdistinguish the elements l 2 L and a 2 A from the variables P , respectivelyS, taking their values in L, respectively A.A trajectory of the generalized hybrid automaton is then de�ned by a timeevent set E , together with the corresponding time evolution �E , and functionsP : �E ! L; x : �E ! X; S : E ! A; w : �E !W , such that- discrete dynamics : for every time event (t0]; t1]; t2]; : : : ; tm(t)])(P (tk]); x(tk]); S(tk]); w(tk]); P (t(k+1)]); x(t(k+1)])) 2 R;k = 0; 1; : : : ;m(t) � 1Furthermore P (~t) = P (tm(t)]), for all ~t > t up to the next event time t0,as well as P (t00]) = P (tm(t)]), and S(t00]) = S(tm(t)]).(Here it is assumed that for all event times t the multiplicitym(t) is either�nite, or, in casem(t) =1, that limk!1 x(tk]) and limk!1 P (tk]) bothexist, in which case we denote these limits by x(tm(t)]), respectivelyP (tm(t)]).)- continuous dynamics : for all event times t1; t2 such that the interval(t1; t2) has zero intersection with ET the function w(t) is piecewise con-tinuous and the function x is continuous and piecewise di�erentiable on

1.2. Towards a de�nition of hybrid systems 17(t1; t2) and satis�esFl(t1)(x(t); _x(t); w(t)) = 0; for almost all t 2 (t1; t2)x(t+1 ) = x(tm(t1)]1 )x(t�2 ) = x(t0]2 )whereas(P (tm(t)]1 ); x(t); S(tm(t)]1 ); w(t); P (tm(t)]1 ); x(t)) 2 R;for all t 2 (t1; t2):(Here, as before, x(t+1 ) := limt#t1 x(t) and x(t�2 ) := limt"t2 x(t).)The above formalization of time evolutions of trajectories of hybrid systemsalso enables us to state a behavioral de�nition of a hybrid system, as follows.De�nition 1.2.6 (Hybrid behavior). Let W = Rq be a continuous com-munication space and let A be a �nite communication space. De�ne the uni-versum U of all possible trajectories given by triples (�E ; a; w), with �E thetime evolution corresponding to some time event set E , and functionsS : E ! Aw : �E !Wwith the property that at all event times t 2 ET with �nite multiplicity m(t)the discrete variables satisfy S(tm(t)]) = S((t0)0]) where t0 is the subsequentevent time. A hybrid system with continuous variables taking values inW anddiscrete variables taking values in A is now de�ned to be a subset B of U , andis denoted by (W;A;B).For many purposes one will actually adapt the above de�nition by shrinkingthe universum U . In particular one may want to restrict to functions w whichhave some regularity properties like continuity or (piecewise) di�erentiability.Furthermore, one may wish to restrict the admissible sets of event times ET .An easy choice is to restrict to sets ET consisting only of isolated points havingno accumulation points. This excludes however examples like the bouncingball in the next chapter (Subsection 2.2.3). Allowing for accumulation points,on the other hand, creates some problems for the compositionality of hybridbehaviors (because the accumulation points are removed from the continuoustime axis). We come back to these issues later on.Remark 1.2.7. The projection of the hybrid behavior on the behav-ior of the discrete variables S is is given by the values taken bythe discrete variables at the time events, that is, on the set oftime instants : : : ; t0]1 ; t]1; t]]1 ; : : : ; tm(t1)]1 ; t0]2 ; t]2; t]]2 ; : : : ; tm(t2)]2 ; t0]3 ; t]3; : : : , where

18 Chapter 1. Modeling of hybrid systems: : : ; t1; t2; t3; : : : are event times. Moreover, if m(ti) is �nite, then S(t0]i+1) =S(tm(ti)]i ). This de�nes the behavior of a discrete-event system.Note, however, that in general the set of \discrete time in-stants" : : : ; t0]1 ; t]1; t]]1 ; : : : ; tm(t1)]1 ; t0]2 ; t]2; t]]2 ; : : : ; tm(t2)]2 ; t0]3 ; t]3; : : : of this result-ing discrete-event system may have a very complicated structure.Analogously, the projection of the hybrid behavior on the behavior of thecontinuous variables de�nes a continuous-time behavior de�ned on the timeaxis R minus the event times and their accumulation points.1.2.6 Event- ow formulasIn this section we provide an alternative framework for modeling hybrid sys-tems, which is equation-based and which is therefore in some sense closer to theusual modeling frameworks for continuous systems than to the graph-relatedrepresentations that are often used for discrete systems. Guards, invariantsand discrete transitions are summarized in De�nition 1.2.5 in one abstract setR. Such a formulation has the advantage of being general, but in practice theset R will usually be described by equations (taken in a general sense to includealso inequalities). Adjoining these equations to the usual sets of di�erentialand algebraic equations leads to a description in terms of what we shall callevent- ow formulas. Such an equation-based framework may be an attractiveway of modeling hybrid systems with a substantial continuous-time dynamics;see also the discussion in Subsection 1.2.8.The style of description in the methodology of event- ow formulas is some-what similar to the way in which \sentences" are described in model theory.We begin by listing the types of variables that may occur. A precise semanticswill be given later, but we already indicate the intended meaning of the sym-bols in order to facilitate the exposition. We use the word \variable" belowto denote symbols that will be subject to evaluation at time points t in thesemantics to be given below. We consider expressions formed from variablesof the following types:- continuous state variables, denoted by x1; x2; : : : ; xn- discrete state variables, denoted by P1; P2; : : : ; Pk- continuous communication variables, denoted by w1; w2; : : : ; wq- discrete communication variables, denoted by S1; S2; : : : ; Sr.The vector (x1; : : : ; xk) will be abbreviated by x, and a similar notationalconvention will be used for the other collections of variables. In the semantics,we will associate to the symbol x a vector of real-valued functions of timetaking values in a space X . Likewise, the continuous communication variablesw will take values in a space W . Each of the discrete state variables Pi willtake values in a �nite set Li; the product L := L1 � � � � � Lk is the set of\locations". Finally, the discrete communication variables Si takes values in

1.2. Towards a de�nition of hybrid systems 19an alphabet Ai, and the product A1 � � � � � Ar will be denoted by A. As asimple mnemonic device, the discrete variables are denoted by capital lettersP and S, while the continuous variables x and w are in lowercase. As notedbefore, the notation also serves to emphasize the nature of a \discrete variable"taking values in the discrete spaces L and A, rather than being an element ofthese discrete spaces.The communication variables may be used to link several parts of a systemdescription to each other. One may also consider \open" systems in which thebehavior of the communication variables is not completely determined by thesystem description itself; in such cases the communication variables may bethought of as providing a link to the (unmodeled) outside world.To the symbols introduced above we associate certain other symbols,namely:- for each continuous state variable xi, there is also a variable _xi (thederivative of xi)- for each continuous state variable xi and each discrete state variable Pithere are variables x]i and P ]i (the next value).We consider expressions in all the above mentioned variables, which areBoolean combinations of what we call ow clauses and event clauses . Forour purposes it seems enough to consider ow clauses which are either of theform �( _x; x; w; P ) = 0 (1.6)(equality type) or of the form�(x;w; P ) � 0 (1.7)(inequality type), where in both cases � is a real-valued function de�ned onthe appropriate domain. Furthermore, we consider event clauses which are ofthe form�(x]; P ]; x; P; S) � 0 (1.8)where again � is a real-valued function de�ned on the appropriate domain.The following de�nition expresses the notion that at each time the systemsthat we consider are subject either to a ow or to an event.De�nition 1.2.8. An event- ow formula, or EFF, is a Boolean formula whoseterms are clauses, and which can be written in the form F _ E where F is aBoolean combination of ow clauses and E is a Boolean combination of eventclauses.Remark 1.2.9. Comparing with the de�nition of a generalized hybrid au-tomaton (De�nition 1.2.5), we see that the ow clauses determine thedi�erential-algebraic equations describing the activities together with \a part"

20 Chapter 1. Modeling of hybrid systemsof the subset R (namely, the part involved in the speci�cation of the continu-ous dynamics), while the event clauses determine the part of R specifying thediscrete dynamics.Remark 1.2.10. We shall freely use alternative notations in cases when it isclear how these can be �tted in the above framework. For instance, we writedown equalities in event conditions even when the above formulation givesonly inequalities, since an equality can be constructed from two inequalities.Also, when for instance P is a discrete variable that may take the values onand off, we write \P = on" rather than �rst de�ning a function � from thetwo-element set fon; offg to R that takes the value 0 on on and 1 on off, andthen writing \�(P ) = 0".Remark 1.2.11. For events with multiplicity 1 it is often notationally easierto use the variables x�i (the left-hand limit) and x+i (the right-hand limit),and analogously P� and P+, to express the values before and after the eventhas taken place. This means that we also use instead of (1.8) event clauses ofthe form�(x+; P+; x�; P�; S) � 0: (1.9)Remark 1.2.12. To further lighten the notation we shall often use a commato represent conjunction, which is a standard convention actually. We use avertical bar to denote disjunction between successive lines, so that������ clause1clause2is read asclause1 _ clause2:We shall also use indexed disjunctions, writing \ji2f1;:::;kgclausei" rather than\clause1 _ � � � _ clausek".For the description of complex systems, it is essential that a compositionoperation is available which makes it possible to combine subsystems intolarger systems. For ow conditions, such a composition may be based simplyon conjunction (logical \and"), to express the intuition that the time we aredealing with here is physical time so that it should be common to all sub-systems. Things are di�erent however in the case of event conditions. If anevent occurs in one subsystem, there are not necessarily events in all othersubsystems; or it may happen that at the same physical time instant there areunrelated events, perhaps of di�erent multiplicities, in several subsystems. Itis therefore useful to introduce the notion of an \empty event", which is de-�ned as follows. All of the alphabets Ai are extended with an element blank

1.2. Towards a de�nition of hybrid systems 21that is di�erent from the existing elements; the interpretation of this value is\no signal". The empty event is given by the clausex+ = x�; P+ = P�; w+ = w�; S = blank: (1.10)We now consider the composition of a number of EFFs. So we start with anumber of event- ow formulas Fi _ Ei (i = 1; : : : ; `), which are thought of asdescriptions of subsystems. All of these subsystems have their own collectionsof symbols which however need not be disjoint. The disjunction of Ei withthe empty event will be denoted by E0i. The parallel composition (or justcomposition) of the subsystems given by Fi _ Ei is now de�ned as the EFF(F1 _ E1) jj (F2 _E2) jj � � � jj (F` _ E`) :== (F1 ^ F2 ^ � � � ^ F`) _ (E01 ^ E02 ^ � � � ^ E 0): (1.11)In this description of composition, communication between subsystems maytake place by shared variables as well as by shared actions.Event- ow formulas may be used for describing hybrid systems in a similarway as di�erential equations are used for describing smooth dynamical systems.As in the latter case, an exact interpretation of the equations requires theconcept of a solution. In continuous systems, a choice has to be made hereas to what function space will be used. In the hybrid system context, we stillhave this question but we also face a few more: in particular, to what extentwill event times be allowed to accumulate, and may multiple events occur atthe same time instant? Already in the case of smooth dynamical systems,there is no such thing as a \correct" answer to the question to what space thesolutions of a given di�erential equation should belong; although some spacesare more popular than others, there is no unique choice that is good for allpurposes and so in practice choices may vary depending on context. Thereis no reason to expect that the situation will be di�erent for hybrid systems.We list a few choices that may turn out to be useful. In each case, we usea continuous state space X , a continuous communication space W , a discretestate space L, and a discrete communication space A. It is convenient to usesome topological concepts.Recall that a \time evolution" is a set of the form�E = (R n ET ) [ Ewith E = [t2ET f(t0]; t1]; : : : ; tm(t)])gwhere ET is a subset of R and m is a function from ET to N [ f1g. We saythat the time evolution �E and the set of time events E are speci�ed by thepair (ET ;m). In order to ease the exposition we restrict throughout to sets ETwhich are closed and nowhere dense subsets of R. As before, we sometimeswrite t]; t]]; : : : for t1]; t2]; : : : . Furthermore, instead of t0] and tm(t)] we alsosometimes write t� and t+ respectively. Sets of the form (t� �; t)[ft�g (with

22 Chapter 1. Modeling of hybrid systemst 2 ET and � > 0) will be written as (t � �; t�], and likewise we will write[t+; t+ �) instead of (t; t+ �) [ ft+g. A time evolution �E speci�ed by a pair(ET ;m) is equipped with a topology generated by subsets of the following fourforms: (t1; t2) � RnET ; (t��; t�] with t 2 ET and (t��; t) � RnET ; [t+; t+�)with t 2 ET and (t; t+ �) � R n ET ; ftj]g with t 2 ET and 0 < j < m(t). (Forthe fundamental mathematical notion of \topology", see for instance [79].)The state spaces X and L and the communication spaces W and A are giventheir usual topologies; in the case of the discrete spaces this means the discretetopology, so that all points of these spaces are viewed as being isolated.We now �rst de�ne a space of trajectories that is about as large as one canhave if one wants to be able to give a meaning to the expressions in an EFF.De�nition 1.2.13. The space Z/1/C1/L1loc consists of tuples (�E ; x; w; P; S),where- �E is a time evolution speci�ed by a pair (ET ;m)- x is a continuous function from �E to X which is continuously di�eren-tiable on R n ET- w is a locally integrable function from R n ET to W having lefthand andrighthand limits at all points of ET- P is a continuous function from �E to L- S is a function from E to A.Remark 1.2.14. Since L is endowed with the discrete topology, continuity ofP means that P is constant outside the event times.Remark 1.2.15. The letter Z (for Zeno) refers to the fact that in the de�-nition the set of event times ET is allowed to have accumulation points; forinstance ET = f 1n j n 2 Z n f0gg[ f0g, or even the Cantor set, could be sets ofevent times. Nevertheless, since ET is assumed to be closed, these accumula-tion points are necessarily elements of ET . (If certain accumulation points arenot in ET , e.g. if we would exclude the accumulation point f0g from the aboveexample of ET , then problems come up in de�ning the continuous dynamicsstarting at such an accumulation point.) The symbol 1 indicates that eventtimes can be of arbitrarily high or even in�nite multiplicity. The symbols C1and L1loc indicate the degree of smoothness that is required for the trajectoriesof the continuous state variables and the continuous communication variableson the open set R n ET . The requirement concerning the existence of lefthandand righthand limits at event times (for the state variables this follows fromthe continuity requirement) is not made explicit in the notation since we willalways impose such a condition.Remark 1.2.16. Note that the de�nition allows the communication variablesw to jump at event times, i. e. we do not necessarily have w(t�) = w(t+) fort 2 ET .

1.2. Towards a de�nition of hybrid systems 23The space we have just introduced is very general; however, we shall oftenwant to work with smaller and more manageable spaces. An example of sucha space is the following.De�nition 1.2.17. The space NZ/1/C1=0/C0 consists of tuples(�E ; x; w; P; S), where- �E is a time evolution speci�ed by a pair (ET ;m) where ET is a set ofisolated points in R, and m(t) = 1 for all t 2 ET- x is a continuous function from �E to X which is continuously di�eren-tiable on R n ET and which satis�es x(t+) = x(t�) for all t 2 ET- w is a continuous function from R n ET to W- P is a continuous function from �E to L- S is a function from E to A.Remark 1.2.18. The acronym NZ in the above notation stands for non-Zeno;it refers to the condition that the points of E are isolated, i. e. there are noaccumulation points. The number 1 indicates that events have multiplicityone so there is no necessity to de�ne intermediate states. The notation C1=0indicates that the continuous state variables are di�erentiable between eventsand continuous across events. Note that in this solution concept discontinuitiesin the external variables w (for instance in the input u) necessarily correspondto events.We also de�ne a third space, which is in some aspects more restricted andin other aspects more general than the previous one. The abbreviation RZused below stands for \right-Zeno".De�nition 1.2.19. The space RZ/1/C1=0 consists of tuples (�E ; x; P ), where- �E is a time evolution speci�ed by a pair (ET ;m) where ET is a set ofright isolated points in R, i. e. for every t 2 ET there is a � > 0 such that(t; t+ �) \ ET = ;, and m(t) = 1 for all t 2 ET- x is a continuous function from �E to X which is continuously di�eren-tiable on R n ET- P is a continuous function from �E to L.Remark 1.2.20. This function class incorporates no communication vari-ables, so as a space for system speci�cation it is suitable only for \monolithic"systems (i. e. systems that are described as one closed whole, without use ofsubsystems and without communication to an outside world). The \bouncingball" discussed in Subsection 2.2.3 is an example of a system that has solu-tions in the space RZ/1/C1=0; the analogous space NZ/1/C1=0 would not besuitable for this example.

24 Chapter 1. Modeling of hybrid systemsVarious other function spaces might be de�ned. Since any attempt atcompleteness would be futile, we do not list any further examples, but weshall feel free to use variants below.At this point we can discuss the notion of solution for EFFs. The notionof solution is based on the evaluation of the elementary clauses in an EFFat speci�c time points. To be precise, a clause of the form �( _x; x; w; P ) = 0will be said to evaluate to true for an element of Z/1/C1/L1loc at a timet 2 R n ET if �( _x(t); x(t); w(t); P (t)) = 0:Likewise, a clause �(x;w; P ) � 0 evaluates to true at t 2 R n ET if�(x(t); w(t); P (t)) � 0:An event clause �(x]; P ]; x; P; S) � 0 evaluates to true at a time tj] 2 E ifeither j < m(t) and�(x(t(j+1)]); P (t(j+1)]); x(tj]); P (tj]); S(tj])) � 0;or j = m(t). Clearly, when clauses have well-de�ned truth values then anyBoolean combination also has a well-de�ned truth value. After these prepara-tions, the notion of solution can be de�ned as follows.De�nition 1.2.21. An element (�E ; x; w; P; S) of Z/1/C1/L1loc is said to bea solution of a given event- ow formula if the ow condition evaluates to truefor all t 2 R n ET and the event condition evaluates to true for all tj] 2 E .Similar de�nitions can be formulated for other time/trajectory spaces. Thefact that in principle di�erent de�nitions have to be given for di�erent functionspaces is already common in the theory of ordinary di�erential equations.Remark 1.2.22. Sometimes a system description can be considerably sim-pli�ed by using what might be called a persistent-mode convention, which en-forces that mode changes will occur only when they are necessary to preventviolation of ow conditions. Typically the involved conditions are expressedas inequality constraints on the continuous state variables. In the mechanicsliterature the persistent-mode convention is sometimes known as \Kilmister'sprinciple" [86], [31]. Formally, a PMC solution for a given event- ow formulais a solution having the property that, for each t0, there is no solution of thesame EFF that coincides with the given solution for t < t0, that is de�ned fort < t1 for some t1 > t0, and that has no event at t0.1.2.7 Simulation of hybrid systemsAfter specifying a complete model for a hybrid system, that is, a syntacticallycorrect model together with a univocal semantics, it can be used for analysis,simulation and control.For simulation purposes it is natural to require that the hybrid model underconsideration has unique solutions for every initial state and every external

1.2. Towards a de�nition of hybrid systems 25\input" signal. Such hybrid systems have been called \well-posed", and it isimportant to derive veri�able conditions which ensure well-posedness.Once it has been established that a particular system is well-posed in thesense that trajectories are uniquely de�ned at least on some interval, the nextquestion arises of actually computing the solutions. Although in principle anyconstructive existence proof for solutions might be used as a basis for calcu-lation, the requirements imposed by numerical e�ciency and theoretical rigorare quite di�erent and so methods of computation may di�er from methodsof proof. This is certainly the case for smooth dynamical systems, where forinstance a Picard iteration may be used for a proof of existence (which ob-tains the solution as a limit of a provably converging sequence of functions),whereas for simulation purposes one would typically use a stepping algorithmin which the di�erence between the value of the state vector at some time andits value at the next time step is approximated on the basis of a few computedfunction values. In the same way, the simulation of hybrid systems is not nec-essarily tied to the theory of well-posedness of such systems. Nevertheless thewell-posedness theory does provide a starting point, and the combination ofalgorithms for mode selection and jump determination with standard methodsfor the simulation of smooth dynamics may already lead to workable simula-tion routines. Much remains to be worked out in this area however and in thissubsection we will only give a very brief outline.Basically there are three di�erent approaches to the simulation of hybridsystems, which each may be worked out in many di�erent ways. The threeapproaches may be brie y described as follows (cf. [114]).1. The smoothing method . In this method, one tries to replace the hybridmodel by a smooth model which is in some sense close to it. For instance,diodes in an electrical network may be described as ideal diodes (possibly plussome other elements), which will give rise to regime-switching dynamics, or asstrongly nonlinear resistors, which gives rise to smooth dynamics. Similarly,in a mechanical system with unilateral constraints one might describe colli-sions as instantaneous, and then one must allow jumps in velocities; or onemight describe them in terms of a compression and a decompression phase,and in that case jumps in velocities may be avoided. In a similar way, unilat-eral constraints in an optimization problem may be replaced by penalty func-tions. Taking an extreme point of view, one might argue that even switchesin computer-controlled processes may be described by smooth dynamical sys-tems; indeed, the transistors inside the computer that physically carry out theswitching can be described for instance in terms of the Ebers-Moll di�erentialequations (see e. g. [46, p. 724]). Reasoning in this way, a point could be madethat it is almost always possible to provide a smooth model that is \closer toreality" than a competing hybrid model.Nevertheless, one can easily come up with examples that would be relativelyawkward to describe on the basis of smoothing. If one looks at a bouncingping-pong ball on a at table, then the nonsmooth model comes to mindimmediately since the time during which the ball is in contact with the table

26 Chapter 1. Modeling of hybrid systemsis very short in comparison with the ight phases. In a smooth model onewould be forced to spend considerable e�ort in specifying various parameterswhich in the nonsmooth model are all captured by one dominant and easilyobservable parameter, namely the restitution coe�cient. Indeed, the strengthof a hybrid model as opposed to a smooth model is usually the simplicity ofthe �rst.Obviously the discussion about whether to use smooth or nonsmooth mod-els should be related to the actual purpose of modeling. If a model for abouncing ball is formulated with the aim of predicting the number of bouncesthat will occur, then the simple nonsmooth model with a constant coe�cientof restitution is not of much use, since it predicts an in�nite number of bounces| an answer which is incorrect and, even worse, not informative. At a generallevel of discussion one cannot make an argument for one or the other. But onecan say that at least for certain problems nonsmooth models are more conve-nient than smooth models. Since in this book we are interested in nonsmoothmodeling, we shall further limit the discussion to that case. We shall not evenspend much attention to the interesting questions around the convergence ofsolutions of smooth models to solutions of nonsmooth models.2. The event-tracking method. The most common way of generating trajec-tories of hybrid systems seems to be the one that is based on the followingsequence:(i) simulation of the smooth dynamics within a given mode (discrete state);(ii) event detection;(iii) determination of a new discrete state (new mode);(iv) determination of a new continuous state (re-initialization).The idea is to simulate the motion in some given mode using a time-steppingmethod until an event is detected, either by some external signal (a discreteinput, such as the turning of a switch) or by violation of some constraintson the continuous state. If such an event occurs, a search is made to �ndaccurately the time of the event and the corresponding state values, and thenthe integration is restarted from the new initial time and initial condition inthe \correct" mode; possibly a search has to be performed to �nd the correctmode.In general one should expect that the behavior within a given mode isactually given by a mixture of di�erential and algebraic equations; for instancein the simulation of a sliding mode one has such a situation, cf. Chapter 3. Thenumerical integration of systems of DAEs has received considerable attentionin recent years, see for instance the books by Brenan et al. [28] and by Hairerand Wanner [61] for more information. In the context of hybrid systems,start-up procedures for DAE solvers should receive particular attention sincere-initializations are expected to occur frequently.

1.2. Towards a de�nition of hybrid systems 27Events within hybrid system simulation can be distinguished in what wecalled externally induced events and internally induced events . Externally in-duced events force a change of mode at a certain time known in advance, suchas when switches are turned in an electrical network according to a prede-termined schedule. Internally induced (or state) events are more di�cult tohandle; these are the events that occur for internal reasons, such as when aninequality constraint becomes active. To catch the internally induced events,a hybrid system simulator needs to be equipped with an event detection mod-ule. Such a module will monitor the sign of certain functions of the state tosee if the required inequality constraints are still satis�ed. In the combinationwith a time stepping algorithm for the simulation of continuous dynamics, onehas to take into account that the time at which an event takes place will ingeneral not coincide with one of the grid points that the continuous simula-tor has placed on the time axis. Both the event time itself and the value ofthe continuous state at the time of the event will have to be found by someinterpolation method.The problem of �nding the next discrete state is called the mode selectionproblem. The problem may be easy in some cases, for instance in a systemin which bu�ers are emptied in a �xed order. There are other cases thoughin which the problem can be quite complicated. Consider for instance a pileof boxes whose relative motion is subject to a Coulomb friction law. Supposethat support on one side is suddenly removed so that the pile will tumbleunder the force of gravity. It is nontrivial to determine which boxes will startsliding with respect to each other and which ones will not. In the case of linearcomplementarity systems it will be shown in Chapter 4 how the next mode canbe actually determined by solving an algebraic problem, and one can use thisas a starting point to look for e�cient numerical methods. In cases in whichthe new mode has to be selected partly on the basis of information from thecontinuous state, one is dealing with a nonconstant mapping from a continuousdomain to a discrete domain. Such a mapping can never be continuous andso one will have to live with the fact that in some cases decisions will be verysensitive. In such situations the simulation software should provide a warningto the user, and if it is di�cult to make a de�nitive choice between severalpossibilities perhaps the solver should even work out all reasonable options inparallel.In many hybrid systems the trajectories of continuous variables can beexpected to be continuous functions, and in these cases the problem of re-initialization comes down to determining the value of the continuous stateat the event time so that the simulation of the smooth dynamics in the newregime can start from an initial state that is correct up to the speci�ed tol-erance. In some cases however, such as in mechanical systems subject tounilateral constraints, jumps need to be calculated, see e.g. the examples pro-vided in Chapter 2. Theoretically, the state after the jump should satisfycertain constraints exactly; �nite word length e�ects however will cause smalldeviations in the order of the machine precision. Such deviations may cause

28 Chapter 1. Modeling of hybrid systemsan interaction with the mode selection module; in particular it may appearthat a certain constraint is violated so that a new event is detected. In thisway it may happen that a cycling between di�erent modes occurs (\livelock"),and the simulator does not return to a situation in which motion accordingto some continuous dynamics is generated, so that e�ectively the simulationstops.3. The timestepping method. In a number of papers (see for instance[111, 124, 146] it has been suggested that in fact it may not be necessaryto track events in order to obtain approximate trajectories of hybrid systems.Moreover, such methods have already been used to implement simulators fordemanding applications like the simulation of integrated circuits with thou-sands of transistors [93]. The term \timestepping methods" has been used torefer to methods that not aim to determine event times; we shall use this termas well, even though of course also the event-tracking methods use time dis-cretization. Rather than giving a formal discussion of timestepping methods,let us illustrate the idea in an example.Consider the following system, which is actually a time-reversed version ofan example of Filippov [54, p. 116]. Let a relay system be given by_x1(t) = � sgnx1(t) + 2 sgnx2(t)_x2(t) = �2 sgnx1(t)� sgnx2(t) (1.12)where the signum function (or relay element) sgn is actually not a functionbut a relation (or multi-valued function) speci�ed by(x > 0 ^ sgnx = 1) _ (x < 0 ^ sgnx = �1) _(x = 0 ^ �1 � sgnx � 1): (1.13)The system (1.12) may be described as a piecewise constant system; in eachquadrant of the (x1; x2)-plane the right hand side is a constant vector. Theinterpretation of systems containing relay elements will be further discussedin Chapter 3. In the case of the simple example above it is strongly suggestedthat solutions should be as pictured in Fig. 1.4. The solutions are spiralingtowards the origin, which is an equilibrium point. It can be veri�ed thatddt (jx1(t)j + jx2(t)j) = �2 which means that solutions starting at (x10; x20)cannot stay away from the origin for longer than 12 (jx10j+ jx20j) units of time.However, solutions cannot arrive at the origin without going through an in�nitenumber of mode switches; since these mode switches would have to occur in a�nite time interval, there must be an accumulation of events.Clearly an event-tracking method is in principle not able to carry out sim-ulation across the accumulation point. The simplest �xed-step discretization

1.2. Towards a de�nition of hybrid systems 29

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Figure 1.4: Solutions of Filippov's example (1.12)scheme for (1.12) is the forward Euler schemex1;k+1 � x1;kh = � sgnx1;k + 2 sgnx2;kx2;k+1 � x2;kh = �2 sgnx1;k � sgnx2;k (1.14)where h denotes the size of the time step and the variable xi;k (i = 1; 2) isintended to be an approximation to xi(t) for t = kh. With the interpretation(1.13) of the signum function this discrete-time system is not deterministic,however. An alternative is to use an implicit scheme. The simplest choice ofsuch a scheme is the following:x1;k+1 � x1;kh = � sgnx1;k+1 + 2 sgnx2;k+1x2;k+1 � x2;kh = �2 sgnx1;k+1 � sgnx2;k+1: (1.15)At each step, x1;k and x2;k are given and (1.15) is to be solved for x1;k+1 andx2;k+1. The equations (1.15) may be written as a system of equalities andinequalities by introducing some extra variables. Simplifying notation a bit bywriting simply xi instead of xi;k+1 and x[i for xi;k , we obtain the following setof equations and inequalities:x1 = x[1 � hu1 + 2hu2 (1.16a)x2 = x[2 � 2hu1 � hu2 (1.16b)(x1 > 0 ^ u1 = 1) _ (x1 < 0 ^ u1 = �1) _ (x1 = 0 ^ �1 � u1 � 1)(1.16c)

30 Chapter 1. Modeling of hybrid systems(x2 > 0 ^ u2 = 1) _ (x2 < 0 ^ u2 = �1) _ (x2 = 0 ^ �1 � u2 � 1):(1.16d)This system is to be solved in the unknowns x1, x2, u1, and u2 for arbitrarygiven x[1 and x[2; h is a parameter. It can be veri�ed directly that for eachpositive value of h and for each given (x[1; x[2) the above system has a uniquesolution; alternatively, one may recognize the system (1.16) as an instance ofthe Linear Complementarity Problem of mathematical programming and inferthe same result from general facts about the LCP. In Figure (1.5) we show thepartitioning of the (x[1; x[2) plane that corresponds to the nine possible waysin which the disjunctions in (1.16c{1.16d) can be satis�ed. For instance, thesolution that has x1 = 0 and x2 = 0 is obtained for the values of (x[1; x[2) suchthat the solution (u1; u2) of24 x[1x[2 35+ h24 �1 2�2 �1 3524 u1u2 35 = 24 00 35 (1.17)satis�es ju1j � 1 and ju2j � 1. A simple matrix inversion shows that thishappens when�5h � x[1 + 2x[2 � 5h; �5h � �2x[1 + x[2 � 5h (1.18)which corresponds to the central area in Fig. 1.5. The solution of the dis-cretized system behaves like that of the original continuous system except inthe narrow strips which do not in uence the solution very much, and except inthe central area where the discretized solution jumps to zero whereas the con-tinuous system continues to go through mode changes at a higher and higherpace. Although we do not present a formal proof here, it is plausible from the�gure that, when the step size h tends to zero, the solution of the discretizedsystem converges to the solution of the original system, including the contin-uation of this solution by x(t) = 0 beyond the accumulation of event times.This happens in spite of the fact that the discretized system only goes through�nitely many mode changes. Note also that the explicit scheme (1.14) showsa rather di�erent and much less satisfactory behavior.The discussion of the example suggests that at least in some cases and byusing suitably selected discretization schemes it is possible to get an accurateapproximation of the trajectories of a hybrid systems without tracking events.Obviously there are many questions to be asked, such as under what conditionsit is possible to use a timestepping method, which discretization methods aremost suitable, which orders of convergence can one get, and what can be gainedby using a variable step size rather than a �xed step size. These matters areto a large extent a matter of future research.1.2.8 Representations of hybrid systemsIn general, the quality and e�ectiveness of any mathematical model dependscrucially on the purpose one wants to use it for. This is particularly true for

1.2. Towards a de�nition of hybrid systems 31h

Figure 1.5: Partitioning of the plane induced by (1.15)models of complex systems, and thus for most hybrid systems. Furthermore,mathematical models of the same system, which are exactly equivalent, mayhave very di�erent properties from the user's point of view. Also, the math-ematical model (or language) describing the functioning of the system maynot be the same as the appropriate language formulating the requirementswhich the system is expected to satisfy (the speci�cations). These featuresare particularly present in the �eld of discrete-event systems (or distributed orconcurrent systems) where one �nds in the literature a wealth of di�erent de-scriptions, from process-algebraic formalisms such as CCS, CSP, ACP, LOTOSand logical theories such as temporal logic to graphically oriented approaches(net theories) such as Petri nets and automata, all with their own advantagesand disadvantages.As a result, it is to be expected that mathematically equivalent descriptionsof a hybrid system, that is, di�erent representations of the same hybrid system,have very di�erent properties, depending on the purpose one wants to use itfor. Also, it is clear that certain representations are better suited for treating aspeci�c subclass of the wealth of hybrid systems than others. Finally, since therepresentation formalism often de�nes the starting point for the developmentof tools for automatically checking certain system properties, the resultingalgorithmic properties of di�erent representations may di�er considerably.In conclusion, di�erent representations of hybrid systems have their ownpros and cons, and one cannot hope for a single representation that will besuitable in all cases and for all purposes. Let us brie y and tentatively discuss

32 Chapter 1. Modeling of hybrid systemssome of the expected merits of the hybrid system representations we have seenso far.It may seem clear that the behavioral de�nition of a hybrid system (Def-inition 1.2.6) is useful from a conceptual point of view and for theoreticalpurposes, but for most other purposes will not yield a convenient descrip-tion. Instead, in order to carry out algorithms, one will usually need a moremanageable and compact hybrid system representation.De�nitions 1.2.3 and 1.2.5 of a (generalized) hybrid automaton do provideworkable representations of hybrid systems for various aims. First of all, theyo�er a clear picture of hybrid dynamics, which is very useful for expositionand theoretical analysis. A favorable feature of the hybrid automaton modelis that the semantics of the model is quite explicit, as we have seen above.Furthermore, for a certain type of hybrid systems and for certain applications,the hybrid automaton representation can be quite e�ective.Nevertheless, a drawback of the hybrid automaton representation is itstendency to become rather complicated, as we shall see already in some of theexamples provided in the next chapter. This is foremost due to the fact thatin the hybrid automaton model it is necessary to specify all the locations andall the transitions from one location to another, together with all their guardsand jumps (or to completely specify the subset R of De�nition 1.2.5). If thenumber of locations grows, this usually becomes an enormous and error-pronetask. Other related types of (graphical) representations of hybrid systemsthat have been proposed in the literature, such as di�erential (or dynamicallycolored) Petri nets, may be more e�cient than the hybrid automaton modelin certain cases but have similar features.For hybrid systems arising in a \physical domain" it seems natural to userepresentation formalisms such as event- ow formulas, which are closer to �rstprinciples physical modeling. First principles modeling of dynamical systemsalmost invariably leads to sets of equations, di�erential or algebraic. Further-more, the hybrid nature of such systems is usually in �rst instance describedby \if-then" or \either-or"statements, in the sense that in one location of thehybrid system a particular subset of the total set of di�erential and algebraicequations has to be satis�ed, while in another location a di�erent subset ofequations should hold. Thus, while in the (generalized) hybrid automatonmodel the dynamics associated with every location are in principle completelyindependent, in most \physical" examples (as we will see in Chapter 2) the setof equations describing the various activities or modes (continuous dynamicsassociated to the locations) will remain almost the same, replacing one or moreequations by some others.Seen from this perspective, the (generalized) hybrid automaton model (andother similar descriptions of hybrid systems) may be quite far from the kind ofmodel one obtains from physical �rst principles modeling, and the translationof the modeling information provided by equations, inequalities and logicalstatements into a complete speci�cation of all the locations of the hybrid au-tomaton together with all the possible discrete transitions and the complete

1.2. Towards a de�nition of hybrid systems 33continuous-time dynamics of every location may be a very tedious operationfor the user. This becomes especially clear in an object-oriented modeling ap-proach, where the interconnection (or composition) of hybrid automata mayeasily lead to a rapid growth in number of locations, and a rather elaborate(re-)speci�cation of the resulting hybrid automaton model obtained by inter-connection. Thus from the user's point of view an interesting alternative fore�ciently specifying \physical" hybrid systems is to look for possibilities ofspecifying such systems primarily by means of equations, as in the frameworkof event- ow formulas. The setting of event- ow formulas is close to that ofsimulation languages such as Modelicatm [50]. Some of the modeling con-structs in Modelica relating to hybrid systems do in fact have the form ofevent- ow formulas. Synchronous languages like LUSTRE [62] and SIGNAL[19] are also related, be it more distantly since these languages operate in dis-crete time; see [18] for an approach to general hybrid systems inspired by theSIGNAL language.The formalism of event- ow formulas results in rather implicit representa-tions of a hybrid system, as opposed to the almost completely explicit repre-sentations provided by the (generalized) hybrid automaton model. The pricethat has to be paid for the use of more implicit representations is that someof the problems in specifying the hybrid system are shifted to the de�nition ofits solutions (the semantics).Within the framework of event- ow formulas one still strives for completespeci�cations of the hybrid system under consideration. In some examples(e.g. the two-carts example, the power-converter example and the variable-structure systems example in Chapter 2) the initial description of the hybridsystem obtained from �rst principles modeling is incomplete, especially withregard to the speci�cation of the discrete dynamics. In fact, one would liketo automatically generate a complete event- ow formula description based onthis initial, incomplete, description, together with some additional information,like the assumption of elastic or non-elastic collisions in Subsection 2.2.9. InChapter 4 we will in fact work out such a framework for a special class ofhybrid systems, called complementarity hybrid systems (including some of theexamples given in Chapter 2).A wealth of di�erent formalisms for describing hybrid systems have beenproposed and are beginning to emerge in the literature. Most of them are ex-tensions of formalisms for describing concurrent systems (extended durationcalculus, hybrid CSP, hybrid state charts, TLA+, Z and duration calculus,VDM++, etc.), and are e�cient only for relatively simple continuous dynam-ics, such as clock time evolution ( _t = 1 or _t = c, where c is some constant) orcontinuous dynamics which can be reasonably approximated in this way. Froma general point of view it seems natural to try to combine process-algebraic for-malisms with the description of continuous dynamics by di�erential-algebraicequations; but no general theory has emerged so far in this direction.

34 Chapter 1. Modeling of hybrid systems1.3 Notes and References for Chapter 1A broad view on the research activities in the area of hybrid systems during thelast decade can be obtained from the conference proceedings [58], [5], [3], [100],[6], [137], [153], as well as the journal special issues [7] and [116]. Needless tosay that many valuable aspects and/or approaches covered in the literaturewill not be addressed in the presentation of hybrid systems in this text. Theterminology \hybrid systems" for this class of systems seems to have been �rstused by Witsenhausen [159]. A recent introduction to various approaches inthe theory of concurrent processes , in particular CSP, can be found in [133];see this book for further references. For temporal logic we refer to [104] and[105]. For an interesting discussion on the similarities and di�erences betweenthe view points of on the one hand computer science and discrete event systems(automata) and on the other hand systems and control theory we refer to [101].There are several useful approaches to hybrid systems that we have not dis-cussed here. Often these approaches have been developed with an eye towardsspeci�c applications or techniques. We mention two directions in particular.In the study of discrete-event systems, Petri nets enjoy great popularity be-cause many situations can be modeled much more e�ciently by a Petri netthan by a �nite automaton with no special structure. For proposals concerningextensions of Petri nets with continuous dynamics, see for instance [43, 119].In contrast, the approach based on mixed logical dynamical (MLD) systemsintroduced in [16] is a discrete extension of a continuous framework. In thisapproach a class of hybrid systems is described by linear dynamic equationssubject to linear inequalities, on the basis of the correspondence that can beconstructed between propositional logic and linear inequalities in real and in-teger variables (see e.g. [157]).

Chapter 2Examples of hybriddynamical systems2.1 IntroductionIn this chapter we treat various (rather simple) examples of hybrid systemsfrom di�erent application areas, with the aim of illustrating the notions ofhybrid models and dynamics discussed in the previous chapter. Some of theexamples will also return in the developments of the following chapters.In order to formalize the examples as hybrid systems we primarily use thenotion of a hybrid automaton (De�nition 1.2.3), or the framework of event- owformulas as introduced in Subsection 1.2.6 of the previous chapter. A numberof notational conventions will be used to facilitate the presentation of event- ow formulas; these have partly already been mentioned in Subsection 1.2.6.A comma is used to indicate logical conjunction between several expressionson one line, a vertical bar is used to indicate logical disjunction between severalexpressions on one line or between successive lines, and a left curly bracket isused to indicate logical conjunction between successive lines. Furthermore, kindicates parallel composition between two subsystems. The symbols associ-ated with a given subsystem are not listed explicitly but are understood asbeing the symbols that occur in the Boolean expressions in the EFF for thatsubsystem. Variables are understood to be continuous across events by de-fault, so we do not explicitly write conditions of the type x+i = x�i ; likewise,conditions of the form S = blank are not written explicitly. Also the emptyevent that goes with each subsystem is not written explicitly.2.2 Examples2.2.1 HysteresisConsider a control system with a hysteresis element in the feedback loop (cf.[26]): _x = H(x) + u (2.1)35

36 Chapter 2. Examples of hybrid dynamical systems1Hx!�1

�� �Figure 2.1: Hysteresiswhere the multi-valued function H is shown in Figure 2.1. Note that thissystem is not just a di�erential equation whose right-hand side is piecewisecontinuous. There is \memory" in the system, which a�ects the right-handside of the di�erential equation. Indeed, the hysteresis function H has anautomaton naturally associated to it, and the system (2.1) can be formalizedas the hybrid automaton depicted in Figure 2.2.H = 1x � �Loc. inv x � ��H = �1_x = 1 + u _x = �1 + uGuard: x � � Loc.inv.Guard: x � �Figure 2.2: Control system with hysteresis as a hybrid automatonIn Figure 2.2 the location invariants are written inside the two circles rep-resenting the two locations, and the transitions (events) are labeled with theirguards. Note that this is a hybrid system involving internally induced switch-ings, but no jumps.If we follow the \persistent-mode convention" of Remark 1.2.22, then anevent- ow formula for this system is simply given by������ _x = 1 + u; x � �_x = �1 + u; x � ��: (2.2)

2.2. Examples 372.2.2 Manual transmissionConsider a simple model of a manual transmission [30]:_x1 = x2_x2 = �a:x2 + u1 + vwhere v is the gear shift position v 2 f1; 2; 3; 4g, u is the acceleration anda is a parameter of the system. Clearly, this is a hybrid system having fourlocations and a two-dimensional continuous state, with controlled transitions(switchings) and no jumps.2.2.3 Bouncing ballConsider a ball bouncing on a table; the bounces are modeled as being in-stantaneous, with restitution coe�cient e assumed to be in the open interval(0; 1). There are no discrete variables (there is only one location), and thereis one continuous variable, denoted by q; this variable indicates the distancebetween the table and the ball.In the hybrid automaton model of this system the system switches from thesingle location back to the same location while an (autonomous) jump occursin the continuous state given by the position q and the velocity _q, since thevelocity changes at an event (impact) time t from _q(t0]) into _q(t]) = �e _q(t0]).The guard of this transition (event) is given by q = 0, _q � 0.The dynamics of the system can be summarized by the di�erential equation(after normalization of all constants)�q = �1 (2.3)if q > 0, together with the discrete transition (impact rule) at an event time �_q(� ]) = �e _q(�0]); (2.4)which occurs if q = 0 and _q � 0.Furthermore the connection between the dynamics at \ordinary" time in-stants and the discrete transition at an event time � is provided by the com-patibility conditionslimt"� q(t) = q(�0]) = q(� ]) = limt#� q(t)limt"� _q(t) = _q(�0])_q(� ]) = limt#� _q(t): (2.5)Using the convention that state variables are continuous across events un-less indicated otherwise, one may write equations (2.3) and (2.4) in an alter-native and more compact form as the event- ow formula (in the continuous

38 Chapter 2. Examples of hybrid dynamical systemsstate vector x = (q; _q))������ q � 0; �q = �1q = 0; _q+ = �e _q�: (2.6)It is clear that the system can be consistently initialized by prescribing q(0�)and _q(0�), with q(0�) � 0. In general it can be a nontrivial question for anevent- ow formula to determine the data one has to provide at 0� in order toensure the existence of a unique solution with these initial data. Prescribingdata at 0� rather than at 0 allows 0 to be an event time.In this example, event times must actually have accumulation points; forinstance if we set q(0�) = 0 and _q(0�) = 1, then it is easily veri�ed thatbounces take place at times 2, 2 + 2e, 2 + 2e + 2e2; : : : , so that we have anaccumulation point at 21�e . Nevertheless we are still able to de�ne a solution:ET = f2Pk�1j=0 ej j k 2 Ng [ f 21�egq(t) = ekt� 12 (t� 2Pk�1j=0 ej)2 for t 2 (2Pk�1j=0 ej ; 2Pkj=0 ej);k = 0; 1; 2; : : := 0 for t > 21�e (2.7)(we have used the standard convention that a summation over an empty setproduces zero). One easily veri�es that this is the only piecewise di�erentiablesolution to (2.6) that is continuous in the sense that the left- and right-handlimits of q(t) exist and are equal to each other for all t. Note that in thepresent example the hybrid trajectory can be naturally extended after theaccumulation point t = 2(1 � e)�1 of event times, since both the continuousstate and the discrete state (there is only one!) converge at the accumulationpoint. In the present case the extension is just the zero trajectory, but it iseasy to modify the example in such a way that the extension is more involved.For some examples of hybrid systems having more than one location, wherethe existence of an extension after the accumulation point is more problematic,we refer to [82].2.2.4 Temperature control systemIn a simple model to be used in temperature control of a room (cf. [1]), wehave one continuous variable (room temperature, denoted by �(t), and takingvalues in R) and one discrete variable (status of the heater, denoted by H(t)and taking values in fon; offg). The continuous dynamics in the system maybe described by an equation of the form_x = f(x;w;H); � = g(x) (2.8)

2.2. Examples 39where f(�; �) is a (su�ciently smooth) function of the continuous state x, thediscrete state H , and the continuous external variable w, which may for ex-ample contain the outside temperature. The hybrid automaton model of thissystem is given by two locations with the obvious location invariants, as givenin Figure 2.3. Note that the speci�cation of the guards is crucial for the set ofon� � 20 o�_x=f(x; o�; w)� � 19� � 20� � 19_x=f(x; on; w)

Figure 2.3: Temperature-thermostat system as a hybrid automatontrajectories of the hybrid automaton. For example, if we take the guards tobe � � 19, respectively � � 20, then transition from one location to another ispermitted while the temperature � is between 19 and 20, and the temperature-thermostat system will behave non-deterministically. This speci�cation maybe appropriate in particular when the thermostat is actually a human con-troller, or in other circumstances in which the thermostat takes other factorsinto consideration than just the one that is explicitly speci�ed in the model. Amore detailed speci�cation may not be worth the e�ort for some applications.A more restrictive speci�cation of the guards would be � = 20, respectively� = 19; note however that this is still not quite enough to obtain unique statetrajectories for all input trajectories.In an event- ow description the system is explicitly modeled as a com-position of two subsystems, namely the heater and the thermostat. So wewrite system = heater jj thermostat (2.9)where the subsystem \heater" is given by (2.8) in which H now acts as adiscrete communication variable, and the subsystem \thermostat" is given by������������ � � 20; H = on� � 19; H = off� � 20; H+ = on� � 19; H+ = off: (2.10)If it is desired to specify for instance that the heating can only be turned onwhen the temperature is exactly 19 degrees, then the �rst event condition in

40 Chapter 2. Examples of hybrid dynamical systems(2.10) should be replaced by� = 19; H� = off; H+ = on: (2.11)Alternatively we can impose the \persistent-mode convention" discussed inRemark 1.2.22.2.2.5 Water-level monitorOur next example (taken from [1]) is somewhat more elaborate than the onesthat we discussed before. The example concerns the modeling of a water-levelcontrol system. There are two continuous variables, denoted by y(t) (the waterlevel) and x(t) (time elapsed since last signal was sent by the monitor). Thereare also two discrete variables, denoted by P (t) (the status of the pump, takingvalues in fon; offg) and S(t) (the nature of the signal last sent by the monitor,also taking values in fon; offg). The dynamics of the system is given in [1] asfollows. The water level rises one unit per second when the pump is on andfalls two units per second when the pump is o�. When the water level risesto 10 units, the monitor sends a switch-o� signal, which after a delay of twoseconds results in the pump turning o�. When the water level falls to 5 units,the monitor sends a switch-on signal, which after a delay of again two secondscauses the pump to switch on.There are several ways in which one may write down equations to describethe system, which may be related to various ways in which the controller maybe implemented. For instance the monitor should send a switch-o� signal whenthe water level reaches 10 units and is rising, but not when the level reaches 10units on its way down. This may be implemented by the sign of the derivativeof y, by looking at the status of the pump, or by looking at the signal last sentby the monitor. Under the assumptions of the model these methods are allequivalent in the sense that they produce the same behavior; however there canbe di�erences in robustness with respect to unmodeled e�ects. The solutionproposed in [1] is based on the signal last sent by the monitor. The hybridautomaton model of this system is given in Figure 2.4.For a description by means of event- ow formulas it seems natural to useparallel composition. One has to spell out in which way the monitor knowswhether the water level is rising or falling when one of the critical levels isobserved. Here we shall assume that the monitor remembers which signal ithas last sent. For that purpose the monitor needs to have a discrete statevariable. We can then write the system as follows:system = tank jj pump jj monitor jj delay (2.12a)with tank ������ P = on; _y = 1P = off; _y = �2 (2.12b)

2.2. Examples 410_y = 1y � 10_x = 1x = 2 x = 2

x := 0x := 0y = 10 _x = 1_y = 1x � 21

_x = 1_y = �2y � 52_x = 1_y = �2x � 23 y = 5Figure 2.4: Water-level monitorpump ������ S0 = sw�on; P+ = onS0 = sw�off; P+ = off (2.12c)monitor ������������ y � 10; Q = req�offy � 5; Q = req�ony = 10; Q� = req�on; Q+ = req�off; S = sw�offy = 5; Q� = req�off; Q+ = req�on; S = sw�on(2.12d)delay ������������ D = inactive; _� = 0D 6= inactive; _� = 1; � � 2D+ = S; �+ = 0� = 2; S0 = D�; �+ = 0; D+ = inactive: (2.12e)Remark 2.2.1. When active, the delay needs a clock (implemented by thevariable �) in order to tell when two units of time have passed. When thedelay is inactive it doesn't need the clock, however; therefore it would perhapsbe better to say that during these periods the clock is \nonexistent" ratherthan to give it some arbitrary dynamics. This would require a modi�cation ofthe setting described here in the spirit of the \presences" of [18].

42 Chapter 2. Examples of hybrid dynamical systems2.2.6 Multiple collisionsThe area of mechanical collisions o�ers a wealth of examples for hybrid systemsmodeling, see e.g. [31] and the references quoted therein. We will only considerthe following seemingly simple case, which is sometimes known as Newton'scradle (with three balls). Consider three point masses with unit mass, mov-ing on a straight line. The positions of the three masses will be denoted byq1; q2; q3, with q1 � q2 � q3. When no collisions take place the dynamics ofthe three masses is described by the second-order di�erential equations_qi = vi_vi = Fifor i = 1; 2; 3, with Fi some prespeci�ed force functions (e.g., gravitationalforces).Collisions (events) take place whenever the positions qi and the velocitiesvi satisfy relations of the formqi = qjvi > vjfor some i 2 f1; 2; 3g with j = i+ 1.The usual impact rule for two rigid bodies a and b with masses ma, respec-tively mb, specifying the velocities v]a; v]b after the impact, is given byv]a � v]b = �e(va � va) (2.13a)mav]a +mbv]b = mava +mbvb (2.13b)where e 2 [0; 1] denotes the restitution coe�cient. Of course, the impact rule(2.13) constitutes already an event idealization of the \real" physical collisionphenomenon, which includes a fast dynamical transition phase. Also notethat (2.13b) expresses conservation of momentum for simple impacts. Belowwe shall apply this rule in a situation where multiple impact occurs (as it wasdone in [63]); of course one may debate whether such an extension is \correct".Let us now consider Newton's cradle with three balls, and suppose that atsome given time instant t 2 R we haveq1 = q2 = q3 = 0v1 = 1v2 = v3 = 0: (2.14)How do we model this case with the above impact rule? The problem is thatnot only the left mass is colliding with the middle mass, but that also themiddle mass is in contact with the right mass; we thus encounter a multiplecollision.

2.2. Examples 43There are at least two ways to model this multiple collision based on theimpact rule (2.13) for single collisions, and these two ways lead to di�erentanswers! One option is to regard the middle and the right mass at the momentof collision as a single mass, with mass equal to 2. Application of the impactrule (2.13) then yieldsv+1 = 13 (1� 2e)v+c = 13 (1 + e)where vc denotes the velocity of the (combined) middle and right mass. Fore = 1 (perfectly elastic collision) this yields the outcome v+1 = � 13 ; v+c = 23 ,while for e = 0 (perfectly inelastic collision) we obtain the outcome v+1 = v+c =13 . Another way of modeling the multiple impact based on the single impactrule (2.13) (cf. [63]) is to imagine that the collision of the left mass with themiddle mass takes place just before the collision of the middle mass with theright mass, leading to an event with multiplicity at least equal to 2.For e = 1 this alternative modeling leads to the following di�erent descrip-tion. The impact rule (2.13) specializes for e = 1 to the event-clausev]i = vjv]j = vi:Hence, we obtain for the initial condition (2.14) an event with multiplicity 2,representing the transfer of the velocity v1 = 1 of the �rst mass to the secondand then to the third mass, with the velocities of the �rst and second massbeing equal to zero after the collision. This behavior is de�nitely di�erent fromthe behavior for e = 1 derived above, but seems to be reasonably close, at leastfor small time, to what one observes experimentally for \Newton's cradle".On the other hand, for e = 0 (perfectly inelastic collision), we obtain inthe second approach from (2.13) the event clausev]i = 12(vi + vj) = v]j :With the same initial conditions (2.14) as above, this gives rise to an eventwith multiplicity equal to 1. In fact, we obtain the following distribution ofvelocities at the subsequents stages t0]; t]; t]]; t]]]; ::: of the time event at timet: v1 v2 v3t0] 1 0 0t] 12 12 0t]] 12 14 14

44 Chapter 2. Examples of hybrid dynamical systemst]]] 38 38 14# # #t1] 13 13 13Hence the outcome of this event with multiplicity1 is the same as the outcomederived by the �rst method, with an event of multiplicity 1.Interesting variations of both modeling approaches may be obtained byconsidering e. g. four unit masses, with initial conditionsq1 = q2 = q3 = q4 = 0v1 = v2 = 1v3 = v4 = 0: (2.15)Another interesting issue which may be studied in the context of this exampleconcerns the continuous dependence of solutions on initial conditions. For abrief discussion of this subject in an example with inelastic collisions, see alsoRemark 4.5.2 below.2.2.7 Variable-structure systemConsider a control system described by equations of the form _x(t) =f(x(t); u(t)), where u(t) is the scalar control input. Suppose that a switchingcontrol scheme is employed that uses a state feedback law u(t) = �1(x(t)) whenthe scalar variable y(t) de�ned by y(t) = h(x(t)) is positive and a feedbacku(t) = �2(x(t)) when y(t) is negative. Writing fi(x) = f(x; �i(x)) for i = 1; 2,we obtain the dynamical system_x = f1(x) if h(x) � 0_x = f2(x) if h(x) � 0: (2.16)Such a system is sometimes called a variable-structure system. The preciseinterpretation of the above equations, which are in principle ambiguous sincethere is no requirement that f1(x) = f2(x) when h(x) = 0, will be discussedbrie y here and more extensively in Chapter 3.A variable-structure system can be considered as a hybrid system with twolocations having di�erent activities; the expressions h(x) � 0 and h(x) � 0serve as location invariants. The problem in specifying this hybrid system isto de�ne the trajectories of the system, starting from initial conditions on thesurface h(x) = 0. The combined vector �eld f(x) de�ned by f(x) := f1(x)for h(x) > 0 and f(x) := f2(x) for h(x) < 0 is in general discontinuous onthe switching surface h(x) = 0. Hence the standard theory for existence anduniqueness of solutions of di�erential equations does not apply, and, indeed, it

2.2. Examples 45���������������

���������������x1

Figure 2.5: Collision to an elastic wallis easy to come up with examples exhibiting multiple solutions for the contin-uous state x. Furthermore, it cannot be expected that state trajectories x(t)are di�erentiable at points where the switching surface is crossed, so it wouldbe too much to require that solutions satisfy (2.16) for all t. One possible wayout is to replace the equation (2.16) by the integral formx(t) = x(0) + Z t0 f(x(s))ds (2.17)which doesn't require the trajectory x(�) to be di�erentiable. A solution of(2.17) is called a solution of (2.16) in the sense of Carath�eodory. The interpre-tation (2.17) also obviates the need for specifying the value of f on the surfacefx jh(x) = 0g, at least for cases in which solutions arrive at this surface fromone side and leave it immediately on the other side.As an example consider the following model of an elastic collision, takenfrom [80]. Consider a mass colliding with an elastic wall; the elasticity of thewall is modeled as a (linear) spring-damper system, as shown in Figure 2.5.The system is described as a system with two locations (or modes):mode 0:8>>>>>><>>>>>>:24 _x1_x2 35 = 24 0 10 0 3524 x1x2 35+ 24 01 35uy = h 1 0 i24 x1x2 35 � 0

mode 1:8>>>>>><>>>>>>:24 _x1_x2 35 = 24 0 1�k �d 3524 x1x2 35+ 24 01 35uy = h 1 0 i24 x1x2 35 � 0

46 Chapter 2. Examples of hybrid dynamical systemsNote that for d 6= 0 the overall dynamics is not continuous on the surface y =x1 = 0, so that the standard theory of existence and uniqueness of solutions ofdi�erential equations does not apply. Nevertheless, it can be readily checkedthat the system has unique solutions for every initial condition (as we expect).On the other hand, modi�cation of the equations may easily lead to a hybridsystem exhibiting multiple solutions. Necessary and su�cient conditions foruniqueness of solutions in the sense of Carath�eodory have been derived in[80, 81].If in the general formulation the vector f1(x0) points inside the setfx jh(x) < 0g and the vector f2(x0) points inside fx jh(x) > 0g for a cer-tain x0 on the switching surface h(x) = 0, then clearly there does not exist asolution in the sense of Carath�eodory. For this case another solution concepthas been de�ned by Filippov, by averaging in a certain sense the \chatteringbehavior" around the switching surface fx jh(x) = 0g. From a hybrid systemspoint of view this can be interpreted as the creation of a new location whosecontinuous-time dynamics is given by this averaged dynamics on the switchingsurface. See for further discussion Chapter 3.2.2.8 Supervisor modelThe following model has been proposed for controlling a continuous-time input-state-output system (1.2) by means of an (input-output) �nite automaton (seeDe�nition 1.2.2); see [8, 120, 26] for further discussion. The model consistsof three basic parts: continuous-time plant, �nite control automaton, andinterface. The interface in turn consist of two parts, viz. an analog-to-digital(AD) converter and digital-to-analog (DA) converter. The supervisor modelis illustrated in Figure 2.6. automatonContinuous-time

symbolInterface

measurement controli 2 I symbol

ADu(�) 2 PU

Controllerplant

o 2 ODA

y 2 Y Figure 2.6: Supervisor modelAssociated to the plant are an input space U , a state space X , and an output

2.2. Examples 47space Y , while the controller automaton has a (�nite) input space I , a statespace Q and an output space O. The controller automaton may be given asan input-output automaton as in (1.3)q] = �(q; i)o = �(q; i):The AD converter is given by a map AD : Y �Q! I . The values of this map(the discrete input symbols) are determined by a partition of the output spaceY , which may depend on the current value of the state of the �nite automaton.The DA converter is given by a map DA : O ! PU , where PU denotes theset of piecewise right-continuous input functions for the plant.The dynamics can be described as follows. Assume that the state of theplant is evolving and that the controller automaton is in state q. Then AD(�; q)assigns to output y(t) a symbol from the input alphabet I . When this symbolchanges, the controller automaton carries out the associated state transition,causing a corresponding change in the output symbol o 2 O. Associated tothis symbol is a control input DA(o) that is applied as input to the plant untilthe input symbol of the controller automaton changes again.In the literature the design of the supervisor is often based on a quantization(also sometimes called abstraction) of the continuous plant to a discrete eventsystem. In this case one considers e.g. an appropriate (�xed) partition of thestate space and the output space of the continuous time plant, together with a�xed set of input functions, and one constructs a (not necessarily deterministic)discrete-event system covering the quantized continuous-time dynamics. Theevents are then determined by the crossing of the boundaries de�ned by thepartition of the state space.2.2.9 Two cartsThe logical disjunction (the \or" between propositions) historically has notbeen entirely absent from the study of continuous dynamical systems; in par-ticular, disjunctions arise in the study of mechanical systems with unilateralconstraints.In the simplest case of a one-dimensional constraint, consider a nonnegativeslack variable which is positive when the system is away from the constraintand zero when the system is at the constraint. The dynamics of the systemwill involve the disjunction of two possibilities: the slack variable is zero andthe corresponding constraint force is nonnegative, or the constraint force iszero and the slack variable is nonnegative. This alternative occurs in classicaltextbooks on mechanics such as [125] and [86], and for the static case goes backto Fourier (cf. [92]). For a concrete example, consider the two-carts examplethat was discussed in [138] and [140]. Two carts are connected to each otherand to a wall by springs; the motion of the left cart is constrained by a stop(see Fig. 2.7). It is assumed that the springs are linear, and all constants

48 Chapter 2. Examples of hybrid dynamical systems������

������

Figure 2.7: Two cartsare set equal to 1; moreover, the stop is placed at the equilibrium position ofthe left cart. There are three continuous variables: q1(t), which denotes theposition of the left cart measured with respect to its equilibrium position, sothat q1(t) also serves as a slack variable; q2(t), which is the position of theright cart with respect to its equilibrium position; and �(t), which denotes theconstraint force. The dynamics of the system can be succinctly written by thefollowing event- ow formula, where e 2 [0; 1] is the restitution coe�cient andthe notationx? y :() ������ x = 0y = 0 (2.18)(for scalar variables x and y) is used:��������������8>>><>>>: �q1 = �2q1 + q2 + ��q2 = q1 � q20 � q1?� � 0q1 = 0; q�1 � 0; _q+1 = �e _q�1 : (2.19)Classically the relation 0 � q1?� � 0 is written as q1 � 0, � � 0, andq1� = 0, which indeed comes down to the same thing and avoids the explicituse of disjunctions.The dynamics of the above system may also be given in a more explicitcondition-event form, with the dynamics in each mode given in the form ofordinary di�erential equations, rather than di�erential-algebraic equations asin the mode descriptions that can be derived directly from (2.19). Since thedi�erential equations are linear, they can even be solved explicitly so that onecan obtain a full description as a hybrid system in the sense of [1]. This de-scription however would be much longer than the one given above (see [141]

2.2. Examples 49cy !0�cu "

Figure 2.8: Coulomb friction characteristicwhere part of the description has been worked out). In general, a systemwith k unilateral constraints can be described by k disjunctions as in (2.19),whereas the number of di�erent locations (discrete states) following from theconstraints is 2k. In this sense the description by means of event- ow formu-las represents an exponential saving in description length with respect to aformulation based on enumeration of the discrete states.The trajectories q1(�) may be taken as continuous and piecewise di�eren-tiable, but not as piecewise continuously di�erentiable since we have to allowjumps in the derivative. In particular, if the stop is assumed to be completelyinelastic (e = 0) then the velocity of the left cart will be reduced to zeroimmediately when it hits the block.In the present example, the speci�cation of the jump rule (the event clause)is quite simple. However, in higher-dimensional examples of this type, espe-cially when multiple collisions are involved, a complete a priori speci�cationof the jumps may not be so easy. In fact, in many cases one would like toavoid an a priori complete speci�cation of the event clauses, and an attractivealternative would be to automatically generate the event-clauses on the basisof inequality constraints of the type q � 0, � � 0, and q� = 0 as above, andphysical information concerning the nature of the collisions, such as the resti-tition coe�cient e (see also [31] for a discussion). In Chapter 4 we shall comeback to this issue in the general context of complementarity hybrid systems,including the present example.2.2.10 Coulomb frictionAn element representing Coulomb friction can be constructed from two con-tinuous variables, say y(t) and u(t), which are related in the following way (seeFigure 2.8):

50 Chapter 2. Examples of hybrid dynamical systemsCoulomb ��������� y � 0; u = cy = 0; �c � u � cy � 0; u = �c: (2.20)Here c is some non-negative constant, while y(t) denotes the velocity of amass and u(t) denotes the frictional force applied to this mass, due to theCoulomb friction. The friction element above can be coupled to another systemdescription (consisting for instance of a continuous system with inputs u(t)(frictional forces) and outputs y(t) (velocities) by conjunction. Whether or notthe complete system will have solutions de�ned for arbitrarily large t dependson the nature of the system added. Notice also that it is not natural to assumea priori that the frictional forces u(t) are continuous; imagine for example aheavy mass sliding subject to Coulomb friction along an upwards inclinedplane, which after coming to rest will immediately slide down subject to thereversed frictional force. Hence we cannot a priori require the functions u(t)and y(t) appearing in (2.20) to be continuous. For the case in which the addedsystem is a linear �nite-dimensional time-invariant input-output system of thetype usually studied in control theory, su�cient conditions for well-posedness(in the sense of existence and uniqueness of solutions) have been given in [95];see also Chapter 4. The determination of the location in systems with multipleCoulomb friction elements is a nontrivial problem, cf. [129] and the commentsin [50].In many cases the constant c appearing in the Coulomb friction is a functionof the normal force applied to the moving mass due to an imposed geometricinequality constraint of the type appearing in Subsection 2.2.9. In this way,by combining the characteristics of Subsection 2.2.9 with those of the presentone, we can describe general multi-body systems with geometric inequalityconstraints and multiple Coulomb friction as hybrid systems; see e.g. [129, 31]for a more extensive discussion.2.2.11 Systems with piecewise linear elementsNote that the Coulomb friction characteristic depicted in Figure 2.8 can bealso interpreted as an ideal relay element (without deadzone). In this case,the third mode (or location) corresponding to the vertical segment of thecharacteristic is usually interpreted in the sense of an equivalent control asde�ned by Filippov; see Chapter 3.More general piecewise linear characteristics can be modeled in a similarway. In this way, any dynamical input-state-output system with piecewise lin-ear characteristics in the feedback loop can be represented as a hybrid system,with the locations corresponding to the di�erent segments of the piecewiselinear characteristics. For more information, especially with respect to well-posedness questions we refer to [33] and the references quoted there; see alsoSubsection 4.1.5.

2.2. Examples 512.2.12 Railroad crossingConsider the railroad crossing from [2]. The system can be described as aconjunction of three subsystems, named `train', `gate', and `controller'. Thetrain sends a signal to the controller at least two minutes before it entersthe crossing. Within one minute, the controller then sends a signal to thegate which is then closed within another minute. At most �ve minutes afterit has announced its approach, the train has left the crossing and sends acorresponding signal to the controller. Within one minute the controller thenprovides a raise signal to the gate, which after receiving this signal takes oneto two minutes to revert to the open position. Below a formal descriptionis given in the form of an EFF in which expressions from the propositionalcalculus are freely used. Several variants are possible depending on the preciseinterpretation that one wants to give to the verbal description. The system isnaturally described as a parallel composition of three subsystems:system = train jj gate jj controller (2.21a)with train ������������ _x = 1; fQT = 1 ) x � 5gPT = out; ST = approach; x+ = 0; Q+T = 1x � 2; P�T = out; P+T = inP�T = in; P+T = out; ST = exit; Q+T = 0 (2.21b)gate ���������������

_y = 1; fQG1 = 1 ) y � 1g; fQG2 = 1 ) y � 2gSC = lower; y+ = 0; Q+G1 = 1P�G = open; P+G = closed; Q+G1 = 0SC = raise; y+ = 0; Q+G2 = 1y � 1; P�G = closed; P+G = open; Q+G2 = 0 (2.21c)controller ���������������_z = 1; fQC = 1 ) z � 1gST = approach; z+ = 0; Q+C = 1SC = lower; Q+C = 0ST = exit; z+ = 0; Q+C = 1SC = raise; Q+C = 0: (2.21d)

The system above is nondeterministic. Note how enabling conditions are for-mulated in the event conditions, and enforcing conditions are placed in the ow conditions.

52 Chapter 2. Examples of hybrid dynamical systemsAn important line of research consists in the development of software toolswhich can help in studying safety and liveness properties on the basis of formaldescriptions like the one above for more complicated systems. Often suchproperties can be expressed as reachability properties; for instance, to use theexamples given in [2], in the train-gate-controller system one may want toverify that the sets fPT = in; PG = openg and fPG = closed; y � 10g arenot reachable.2.2.13 Power converterConsider the power converter in Figure 2.9 (cf. [51]). The circuit in Figure 2.9+E RCs = 1 s = 0 ��+ � +�++�

Figure 2.9: Boost circuit with clamping diodeconsists of an inductor L with magnetic ux linkage �L, a capacitor C withelectric charge qC and a resistance load R, together with a diode and an idealswitch, with switch positions s = 1 (switch closed) and s = 0 (switch open).The diode is modeled as an ideal diode with voltage-current characteristic givenby Figure 2.10. The constitutive relation of an ideal diode can be succinctly0 V !

I "Figure 2.10: Ideal diode

2.2. Examples 53expressed as follows:vDiD = 0; vD � 0; iD � 0: (2.22)The circuit is used to obtain a voltage at the resistance load (the outputvoltage) that is higher than the voltage E of the input source; therefore it iscommonly called a step-up converter.Intuitively it is clear that the system can be represented as a hybrid systemwith four locations (or modes), corresponding to the two segments of the diodecharacteristic and the two switch positions. Furthermore, the transitions froma location with open switch to a location with closed switch, and vice versa, arecontrolled (externally induced), while the transitions corresponding to a changefrom one segment of the diode characteristic to another are autonomous.Taking as continuous state (energy) variables the electric charge qC and themagnetic ux �L, and as stored energy the quadratic function 12C q2C + 12L�2Lwe obtain the following dynamical equations of the circuit:24 _qC_�L 35 = 24 � 1R 1� ss� 1 0 3524 qCC�LL 35+ 24 01 35E + 24 siD(s� 1)vD 35 :(2.23)Here s = 0; 1 denotes the switch, E is the voltage of the input source, andiD; vD , are respectively the current through the diode, and the voltage acrossthe ideal diode. The dynamics of the circuit is completely speci�ed by (2.23)together with the switch position (a discrete variable) and the constitutiverelation of the ideal diode given by (2.22).The separate dynamics of the four locations are obtained by substitutingthe following equalities into (2.23).- Location 1 : s = 0; vD = 0- Location 2 : s = 1; iD = 0- Location 3 : s = 0; iD = 0- Location 4 : s = 1; vD = 0This yields for each of the four locations the following continuous dynamics:_qC = � 1L�L � 1RC qC_�L = � 1C qC +E_qC = � 1RC qC_�L = E

54 Chapter 2. Examples of hybrid dynamical systems_qC = � 1RC qC_�L = 0_qC = 0_�L = EIn order to �nd the currently active location, we �rst observe the positionof the switch. For s = 0 we have the locations 1 and 3. The location 1 isdetermined by vD = 0 and iD = �LL � 0. The latter inequality yields thelocation invariant �L � 0. The location 3 is given by iD = �LL = 0 andvD = E � qCC � 0. This yields the location invariants �L = 0 and qC �E � 0.Similarly, if s = 1 then the system is in location 2 for iD = 0 and the voltagevD = � qCC � 0, giving the location invariant qC � 0, and in location 4 ifvD = qCC = 0 and iD � 0, leading to the location invariant qC = 0.Furthermore, it is straightforward (but tedious!) to write down all thetransition guards, and the jump relations. In fact, it can be seen that in\normal operation", that means, if we start from an initial continuous statewith �L � 0 and qC � 0 and | very importantly | with the input voltageE � 0, then no jumps will occur and �L(t) � 0 and qC(t) � 0 for all t > 0.Let us note that one of the two location invariants of location 3, namelyqC � E � 0 explicitly depends on the external continuous variable E. Thusthe example �ts into the generalized hybrid automaton model, but not in thehybrid automaton model of De�nition 1.2.3. Also, note that it makes senseto de�ne the continuous state space of locations 3 and 4 to be given by theone-dimensional spaces f(qC ; �L) j�L = 0g, respectively f(qC ; �L) j qC = 0g,in accordance with the remark made before that in some cases it is natural toallow in the (generalized) hybrid automaton model for di�erent state spacesfor the various locations.Equations (2.23) and (2.22) provide an (incomplete) event- ow formuladescription of the system; incomplete because the event-clauses correspondingto the diode have not been speci�ed. The example demonstrates that the(generalized) hybrid automaton model may be far from an e�cient descriptionof a hybrid system: while the event- ow model given by (2.23) and (2.22)follows immediately from modeling, the hybrid automaton representation ofthis simple example already becomes involved.2.2.14 Constrained pendulumThis example, which has been extensively discussed in [27], is used here toillustrate that the choice of state space variables may be crucial for the com-plexity of the resulting hybrid system description.1 Consider a mathematicalpendulum with length l that hits a pin such that the constrained pendulumhas length lc, cf. Figure 2.11. Taking as continuous state space variables1This point was kindly brought to our attention by P.C. Breedveld.

2.2. Examples 55

������������

������������

����������������

������

�������pin �pin `�`cFigure 2.11: Pendulum constrained by a pinx = (�; v), where v is the angular velocity of the end of the pendulum, weobtain a hybrid system with two locations (unconstrained and constrained).The unconstrained dynamics, valid for � � �pin, is given by_� = 1l v_v = �g sin�� �v (2.24)for some friction coe�cient �. The constrained dynamics, valid for � � �pin,is given by the same equations with l replaced by lc. It is immediately seenthat there are no jumps in the continuous state vector x at the event times.Instead there is only a discontinuity in the righthand side of the di�erentialequation caused by the change from l to lc, or vice versa.On the other hand, if we would take as continuous state space variables �and _�, then at the moment that the swinging pendulum hits the pin, there isa jump in the second state space variable from _� to llc _�, and conversely, if thependulum swings back from the constrained movement to the unconstrainedmovement there is a jump from _� to lcl _�.Clearly the resulting hybrid description is more complex from the hybridsystems point of view than the (equivalent!) description given before. On theother hand, from a physical modeling point of view the occurrence of jumps dueto the \collision" of the rope with the pin is rather natural, while the \smart"choice of the continuous state variables in the �rst description eliminates insome way these jumps.2.2.15 Degenerate Van der Pol oscillatorIn this example (taken from [136]) we indicate that systems described bydi�erential-algebraic equations may exhibit jump features, which could mo-tivate a description as a hybrid system. Consider a degenerate form of the

56 Chapter 2. Examples of hybrid dynamical systemsvan der Pol oscillator consisting of a 1-F capacitor in parallel with a nonlinearresistor with a speci�c characteristic:_v = iv = � 13 i3 + i: (2.25)These equations are interpreted as an implicitly de�ned dynamics on the one-dimensional constraint submanifold C in (v; i) space given byC = f(v; i) j v = �13 i3 + ig:Di�culties in this interpretation arise in the points (�1;� 23 ) and (1; 23 ). Atthese points _v is negative, respectively positive, while the corresponding time-derivative of i in these points is positive, respectively negative. Hence, becauseof the form of the constraint manifold C it is not possible to \integrate" thedynamics from these points in a continuous manner along C.Instead it has been suggested in [136] (see this paper for other relatedreferences) that a suitable interpretation of the dynamics from these singularpoints is given by the folowing jump rules:(�1;� 23 ) ! (2;� 23 )(1; 23 ) ! (�2; 23 ) (2.26)Alternatively, the resulting system can be described as a hybrid system withtwo locations with continuous state spaces both given by C and dynamicsdescribed by (2.25), with location invariants i � �1, respectively i � 1, andjump relations given by (2.26).2.3 Notes and References for Chapter 2The choice of the examples in this chapter clearly re ects the interests and biasof the authors of this text. A wealth of di�erent examples of hybrid systemscan be found in the literature, see in particular the proceedings [58], [5], [3],[100], [6], [137], [153], as well as the journal special issues [7] and [116]. Forfurther examples and references in the context of mechanical problems, anexcellent source is [31].

Chapter 3Variable-structure systemsDiscontinuous dynamical systems have been an object of systematic study inthe former Soviet Union and other Eastern European countries for a long pe-riod starting in the late 1940s. Much of the theory has been developed in closeconnection with control theory, which has provided many motivating exam-ples such as relay controls and bang-bang controls in shortest-time problems.Systematic expositions of the results of this research are available in a num-ber of textbooks, for instance the ones by Andronov et al. [4], Tsypkin [150],Utkin [152], and Filippov [54]. Here we shall not attempt to summarize all thiswork; instead we concentrate on the solution concept for discontinuous dynam-ical systems, and more speci�cally on the so-called \sliding mode". We followmainly Filippov's book. We do not aim for the greatest possible generality; inparticular we limit ourselves to systems with constant parameters.3.1 Discontinuous dynamical systemsA typical example of the type of systems considered by Filippov can be con-structed as follows. Let S0 be a surface in n-dimensional space, by which wemean that S0 is a (n � 1)-dimensional di�erentiable manifold determined asthe null set of a smooth real-valued function � on Rn . The set of all x forwhich �(x) is positive (negative) will be denoted by S+ (S�). Let f+ be acontinuous function de�ned on S+[S0, let f� be a continuous function de�nedon S� [ S0, and let f be the function de�ned on S+ [ S� by f(x) = f+(x)for x 2 S+ and f(x) = f�(x) for x 2 S�. It is not required that the functionsf+ and f� agree on S0, so that in general the function f cannot be extendedto a continuous function de�ned on all of Rn . Now consider the di�erentialequation_x(t) = f(x(t)): (3.1)Here we have a dynamic system whose dynamics changes abruptly when thestate vector crosses the switching surface S0. In a control context, such a sit-uation could occur as a result of a gain scheduling control law which switchesfrom one feedback to another when a certain function of the state variablescrosses a certain threshold. In general it cannot be expected that state tra-jectories are di�erentiable at points where the boundary is crossed, and so it57

58 Chapter 3. Variable-structure systemswould be too much to require the validity of (3.1) for all t. One possible wayout is to replace the equation (3.1) by the integral formx(t) = x(0) + Z t0 f(x(s))ds (3.2)which doesn't require the trajectory x(�) to be di�erentiable. When di�erentialequations are interpreted in this way they are sometimes called Carath�eodoryequations . The interpretation (3.2) also obviates the need for specifying thevalue of f on the surface S0, at least for cases in which solutions arrive at thissurface from one side and leave it immediately on the other side.Whether or not we do have solutions that cross the switching surface instan-taneously depends on the vector �elds determined by f+ and f�. Consider apoint x0 on the switching surface S0. At this point we have two vectors f+(x0)and f�(x0). In terms of these vectors and their relation to the tangent spaceof the surface S0 at the point x0 we can distinguish the following four maincases.(i) Both vectors point inside S+. In this case, state trajectories can onlyarrive at x0 from S�, and will continue in S+. The Carath�eodory inter-pretation is su�cient.(ii) Both vectors point inside S�; this is analogous to case (i).(iii) The vector f+(x0) points inside S+ and the vector f�(x0) points insideS�. In this case x0 cannot be reached by trajectories of (3.1). If x0 istaken as an initial condition, there are two possible solutions to (3.2).(iv) The vector f+(x0) points inside S� and the vector f�(x0) points insideS+. In this case the Carath�eodory interpretation does not give us ausable solution concept.Filippov is mainly concerned with situations in which case (iv) occurs. Itcan be argued that it is physically meaningful to try to develop also a solutionconcept for this case. For instance, it may be that the function f is a simpli�edversion of another function ~f that is not actually discontinuous across theswitching surface, but that changes in a steep gradient from f+ on one side tof� on the other side of S0. The solutions of the di�erential equation _x = ~f(x)will in case (iv) tend to follow the switching surface since they are \pushed"onto S0 from both sides. In another interpretation, suppose that the transitionfrom the regime described by f+ to the one described by f� is caused bya switching controller that monitors the sign of �(x(t)). For a number ofpractical reasons, switching will not occur exactly when �(x(t)) crosses thezero value, but at some nearby instant of time. In case (iv) the result will be a\chattering" behavior in which the system switches quickly from one dynamicsto the other and back again. Also in this way a motion will result which willtake place more or less along the switching surface. So on the one hand thereare good reasons to allow for solutions along the switching surface in cases

3.2. Solution concepts 59of type (iv), on the other hand several di�erent physical mechanisms may beat work, and one can also say that the situation is perhaps not completelyspeci�ed by the two functions f+ on S+ and f� on S�. Therefore Filippovactually discusses several di�erent solution concepts.3.2 Solution conceptsLet us �rst look at what Filippov calls the simplest convex de�nition. Takethe situation of case (iv); so consider a point x0 on the switching surface, andsuppose that the vectors f+(x0) and f�(x0) point inside S� and S+ respec-tively. Since these two vectors are at di�erent sides of the tangent space ofS0 at x0, there must be a convex combination of them which lies in the tan-gent space. Denote the vector obtained in this way by f0(x0). Repeating theconstruction for points x on S0 in a neighborhood of x0, we obtain a functionf0(x) de�ned on S0 at least in a neighborhood of x0, and having the propertythat the vector f0(x) always points in the tangent space of S0 at x. Therefore,the di�erential equation _x = f0(x) can be used to de�ne a motion on S0 whichis called a sliding motion. The concept of solution is now extended to includethis type of motion as well.The second notion of solution discussed by Filippov uses the so-called equiv-alent control method . For the application of this method, the function f issupposed to be of the form f(x; u(x)) where u(x) is a multivalued functionthat is in fact single-valued on S+[S� but that has a range of values U(x) forx 2 S0. The set U(x) is a closed interval. In the situation of case (iv) above,an \equivalent control" ueq(x) is sought for x 2 S0 such that f(x; ueq(x)) istangent to S0 and ueq(x) 2 U(x). The motion given by _x = f(x; ueq(x)), whichis a motion along the sliding surface as in the case of the simplest convex def-inition, is then used to de�ne a notion of solution. The resulting motion maybe quite di�erent from the one produced by the �rst de�nition; see Examples3.2.1 and 3.2.2 below.Filippov also considers a third de�nition. This de�nition again starts fromthe formulation _x = f(x; u(x)) with u(x) 2 U(x), where U(x) is a single pointfor x 2 S+ [ S� and is a closed interval for x 2 S0. For given x0, let F (x0)denote the smallest convex set containing ff(x0; u) j u 2 U(x0)g. We can nowconsider the di�erential inclusion _x(t) 2 F (x(t)). Away from the switchingsurface, this inclusion is in fact a standard di�erential equation since F (x)then consists of only a single point. On the switching surface, the requirement_x(t) 2 F (x(t)) leaves considerable latitude; however, solution trajectories muststill follow the switching manifold in the neighborhood of points where case(iv) applies, because solutions that enter either S+ or S� are not possible.In case f(x; u) depends a�nely on u and the interval U(x0) is [u+; u�] withu+ = limx2S+; x!x0 u(x) and u� de�ned likewise, all solution concepts are thesame. In other cases however, the third de�nition does not uniquely determinethe velocity of the motion along the switching surface. The indeterminismthat is introduced this way may be viewed as a way of avoiding the choice

60 Chapter 3. Variable-structure systemsbetween the other two solution concepts. Such a conservative stance can bewell-motivated in a veri�cation analysis in which one would like to build in adegree of robustness against model misspeci�cation. From a point of view ofsimulation however, one would rather work with uniquely de�ned trajectories.It might be taken as an objection against the equivalent control methodthat it requires the modeler to specify a function f(x; u) and a function u(x)on S+ [ S� together with a closed interval U(x) for x 2 S0; in this way, themodeler is forced into decisions that he or she would perhaps prefer to avoid.The simple convex de�nition only requires the speci�cation of functions f+ onS+ [ S0 and f� on S� [ S0. It may be argued however that assuming thesimple de�nition comes down to making a particular choice for the functionsf(x; u) and u(x) that are used in the equivalent control method. For thispurpose, assume that the continuous functions f+ and f� de�ned on S+ [ S0and S� [ S0 respectively are extended in some arbitrary way to continuousfunctions to all of Rn . If we now de�ne f(x; u) byf(x; u) = 12 (1 + u)f+(x) + 12 (1� u)f�(x) (3.3)then f is continuous as a function of x and u. Furthermore de�ne u(x) =sgn(�(x)), where sgn is the multivalued function de�ned by (1.13). Note thatf(x; u) is a�ne in x. It is easily veri�ed that the equivalent control methodand the third de�nition now both generate solutions that coincide with theones obtained from the simple convex de�nition.To illustrate the di�erence between the equivalent control method and thesimplest convex de�nition, we present the following two examples.Example 3.2.1. This example is taken from [109]. Let a system be given by_x1 = cos �u; _x2 = � sin �u; y = x2; u = sgn y: (3.4)Both for x2 > 0 and for x2 < 0 the right hand side is constant and so thesystem above could be called a \piecewise constant system". The trajectoriesare straight lines impinging on the x1-axis at an angle determined by theparameter � which is chosen from (0; �). Along the x1-axis, a sliding mode ispossible. The equivalent control method applied to (3.4) determines u suchthat _x2 = � sin �u = 0; obviously this requires u = 0 so that the slidingmode is given by _x1 = 1. If one would take the simplest convex de�nition,one would look for a convex combination of the vectors col(cos �;� sin �) andcol(cos �; sin �) such that the second component vanishes. There is one suchconvex combination, namelycol(cos �; 0) = 12col(cos �;� sin �) + 12col(cos �; sin �):In this case the sliding mode is given by _x1 = cos �. The question which ofthe two sliding modes is the \correct" one has no general answer; di�erentapproximations of the relay characteristic may lead to di�erent sliding modes.The third de�nition leads to the di�erential inclusion _x1 2 [cos �; 1].

3.2. Solution concepts 61Let us now consider a smooth approximation as well as a chattering ap-proximation to the sliding mode. In the smooth approximation, we assumethat there is a very quick but continuous transition from the vector �eld onone side of the switching surface to the vector �eld on the other side. Thismay be e�ectuated by replacing the relation u = sgn y by a sigmoid-typefunction, for instanceu = tanh(y=") (3.5)where " is a small positive number. Doing this for (3.4) we get the smoothdynamical system_x1 = cos(� tanhx2); _x2 = � sin(� tanhx2) (3.6)whose trajectories are very similar to those of (3.4) at least outside a small bandalong the switching curve x2 = 0. The system (3.6) has solutions x1(t) = t+c,x2(t) = 0 which satisfy the equations of the sliding mode according to theequivalent control de�nition.Consider next the chattering approximation. Again we choose a smallpositive number ", and we de�ne a \chattering system" by the event- owformula������������ P = up; _x1 = cos �; _x2 = � sin �; x2 � �"P = down; _x1 = cos �; _x2 = sin �; x2 � "x2 = �"; P� = up; P+ = downx2 = "; P� = down; P+ = up: (3.7)The trajectories of this system are exactly the same as those of the originalsystem (3.4) except in a band of width 2" around the switching curve. Forsmall " the system (3.7) has solutions that are close to trajectories of the formx1(t) = t cos � + c, x2(t) = 0. These trajectories satisfy the equations of thesliding mode according to the simplest convex de�nition.It may be noted that (3.6) is not the only possible smooth approximationto (3.4); another possibility is for instance_x1 = cos �; _x2 = � tanh(x2=") sin �: (3.8)The trajectories of this system are arbitrarily close to those of (3.4) outside aband around the switching curve if " is taken small enough. The system (3.8)has solutions of the form x1(t) = t cos � + c, x2(t) = 0 which conform to theequations of the sliding mode according to the simplest convex de�nition.The solution according to the simplest convex de�nition can be obtainedas an equivalent control solution if we replace the equations (3.4) by_x1 = cos �; _x2 = �u sin �; y = x2; u = sgn y: (3.9)Actually in this case the smooth approximation according to the recipe (3.5)leads to the system (3.8).

62 Chapter 3. Variable-structure systemsExample 3.2.2. Consider now the system_x1(t) = �x1(t) + x2(t)� u(t) (3.10)_x2(t) = 2x2(t)(u2(t)� u(t)� 1) (3.11)u(t) = sgnx1(t): (3.12)The system has a sliding mode on the interval x1 = 0, �1 � x2 � 1. Accordingto the simplest convex de�nition, the sliding mode is given by_x2 = �2x22 (3.13)wheras according to the equivalent control method the sliding mode is givenby _x2 = 2x2(x22 � x2 � 1): (3.14)The two dynamics (3.13) and (3.14) are quite di�erent; (3.13) has an unstableequilibrium at 0 whereas (3.14) has two equilibria, of which the one at 12� 12p5is unstable and the one at 0 is stable. In particular, solutions of the system(3.10) in the \simplest convex" interpretation and in the \equivalent control"interpretation that are identical until they reach a point on the x2-axis with12 � 12p5 � x2 � 0 will after this point follow entirely di�erent paths. Theequivalent control solution will be recovered by a smoothing approximationsuch as u = tanh(x1="), whereas other methods that are based on some typeof sampling will follow the solution according to the simplest convex de�nition.3.3 ReformulationsThe generality in the formulation of the equivalent control method may bereduced somewhat without loss of expressive power. As above, let u+ and u�be extended arbitrarily to continuous functions on all of Rn . Introduce a newvariable v, and de�ne a continuous function ~f of the two variables x and v by~f(x; v) = f(x; 12 (1� v)u�(x) + 12 (1 + v)u+(x)):Given that f is continuous as a function of x and u, the function ~f will becontinuous as a function of x and v. Moreover, we have ~f(x; 1) = f(x; u+(x))and ~f(x;�1) = f(x; u�(x)). In the most important special case in which theend points of the interval U(x) for x 2 S0 are u+(x) and u�(x), we can simplyde�ne v(x) = sgn(�(x)) to get the same solutions to _x = ~f(x; v(x)) as onewould get from _x = f(x; u(x)) according to the equivalent control method.In Filippov's treatment, the wordmode is used (in the term \sliding mode")but discontinuous systems are not modeled explicitly as hybrid systems. Byrewriting the equations in terms of event- ow formulas, more emphasis isplaced on the multimodal aspects. In the equivalent control method, weare given a function f(x; u) and a multivalued function u(x). On S+ and

3.4. Systems with many regimes 63S�, u is single-valued; denote the corresponding functions on S+ and S� byu+ and u� respectively. For x 2 S0, u can take values in a closed intervalU(x) =: [u+(x); u�(x)]. It seems reasonable to conjecture (but of course itneeds proof) that, under fairly general circumstances, Filippov's solutions ac-cording to the equivalent control de�nition correspond to the continuous statetraces of the solutions in Z/1/C1=0/C0 of the EFF�������������������������� _x = f(x; u); y = �(x)��������� y > 0; u = u+(x)y < 0; u = u�(x)y = 0; u�(x) � u � u+(x) (3.15)For the solution concept corresponding to the simple convex de�nition onewould rather use the following EFF:�������������������������� _x = 12 (1 + u)f+(x) + 12 (1� u)f�(x); y = �(x)��������� y > 0; u = 1y < 0; u = �1y = 0; �1 � u � 1 (3.16)The three-term disjunction in the parts corresponding to non-event times in-dicates a system with three modes, one corresponding to motion inside S+,another corresponding to motion inside S�, and a sliding mode. In the slidingmode, trajectories must move along the switching manifold characterized byy = 0, and this motion must take place along a vector of the form f(x; u) withu 2 U(x) = [u�(x); u+(x)] or along a suitable convex combination of vectorsf+(x) and f�(x). The motion on S+ and S� follows the di�erential equations_x = f+(x) and _x = f�(x) respectively. All this is expressed in (3.15) and(3.16).It should be emphasized that, although the de�nitions have been designedto avoid some obvious cases of nonexistence of solutions, neither existence noruniqueness of solutions to the type of systems considered in this section isautomatic. We shall come back to this issue below.3.4 Systems with many regimesSo far we have been discussing situations in which there is one surface thatdetermines the various regimes under which the system can evolve. Such asituation occurs for instance when friction at one point in a mechanical systemis modeled by the Coulomb friction law. Of course we can easily have situationsin which there are many points of friction and then the space through which

64 Chapter 3. Variable-structure systemsthe continuous state variables of the system move will be divided in manyparts each with their own dynamical regime. Suppose that we have k surfacesdetermined by functions �1; : : : ; �k from Rn to R. In principle, one should nowallow sliding modes not just along each of the surfaces Si0 := fx j �i(x) = 0gbut also along the intersection Si0 \ Sj0 of two surfaces, which will in generalbe a manifold of dimension n � 2, as well as along the intersection of threesurfaces and so on. To create these sliding modes, one will in general need kindependent control variables, so the given dynamics should be of the form_x = f(x; u1; : : : ; ur)with r � k; moreover, there should be enough freedom in the variables ui(x)when x is in the intersection of several surfaces Si0 to allow motion alongthese intersections. If one wants to ensure uniqueness of solutions, then itis natural to let r be equal to k and also to establish a one-one connectionbetween control variables and surfaces. The existence of such a connection,which ties each input variable ui to a corresponding \output variable" de�nedby yi = �i(x), is anyway natural in applications such as relay systems andCoulomb friction. An event- ow formula may be written down as follows:_x = f(x; u) jji2kf yi = �i(x); (yi > 0; ui = ui+(x)) j(yi < 0; ui = ui�(x)) j (yi = 0; ui�(x) � ui � ui+(x))g (3.17)where k denotes the set f1; : : : ; kg; solutions are sought in a space with continu-ous state trajectories. As above, in case ui+(x) = ui+(x) and ui�(x) = ui�(x)for all i and all x 2 S0, it is possible to rewrite the system in such a way thatthe control variables are related to the output variables �i(x) by the signumfunction sgn. The discontinuous system is then rewritten as a relay system.In the case of several control variables the relation between the equiva-lent control method and the simplest convex de�nition can be more compli-cated than in the case of a single control variable. Consider the example_x = f(x; u1; u2) with ui = sgn(�i(x)) for i = 1; 2. If �1(x) = �2(x) thestate space is actually divided into two parts, and the dynamics is given by_x = f(x; 1; 1) on one side and by _x = f(x;�1;�1) on the other side of thedividing surface. The simple convex de�nition would require a sliding motionalong the surface to be generated by some convex combination of the two vec-tors f(x; 1; 1) and f(x;�1;�1), whereas the equivalent control method wouldallow the motion to be generated by an arbitrary vector of the form f(x; u1; u2)with �1 � u1 � 1 and �1 � u2 � 1. Even in the case in which f dependslinearly on u1 and u2 the allowed motions according to the two de�nitions arein general di�erent. The example is perhaps arti�cial though; in fact with thesimple convex de�nition there is little reason to consider the system as onehaving two control variables. In case the system is interpreted as having twocontrol variables and the equivalent control method is used to de�ne solutions,it is likely that the solutions are not uniquely determined since in the slidingmode there are two control variables available to satisfy only one constraint.

3.5. The Vidale-Wolfe advertizing model 653.5 The Vidale-Wolfe advertizing modelAs an example of a situation in which multimodality arises in a natural way,let us consider the following optimization problem which was suggested in 1957by Vidale and Wolfe as a simple model for supporting decisions on marketingexpenditures. The problem is:maximize Z T0 (x(t)� cu(t))dt (3.18a)subject to _x(t) = �ax(t) + (1� x(t))u(t) (3.18b)x(0) = x0 2 (0; 1) (3.18c)0 � u(t) � 1: (3.18d)The state variable x(t) represents market share, whereas the control variableu(t) represents marketing e�ort, normalized in such a way that saturationoccurs at the value 1. The constant c expresses cost of advertizing, and aindicates the rate at which sales will decrease when no marketing is done. Astandard application of the maximum principle (see for instance [142]) suggeststhe following procedure for �nding candidate optimal solutions. First de�nethe HamiltonianH(x; u; �) = x� cu+ �(�ax+ (1� x)u): (3.19)According to the maximum principle, necessary conditions for optimality aregiven (in the \normal" case) by the equations (in shorthand notation)_x = @H@� ; _� = �@H@x ; x(0) = x0; �(T ) = 0; u = arg maxuH(3.20)where �(�) is an adjoint variable. Since the Hamiltonian is linear in the controlvariable u, maximization over u leads to a relay-like characteristic:��������� u = 0; �c+ �(1� x) < 0u = 1; �c+ �(1� x) > 00 � u � 1; �c+ �(1� x) = 0: (3.21)It will be convenient to introduce a function C byC(x; �) := �c+ �(1� x): (3.22)The relay characteristic is connected to the di�erential equations_x = �ax+ (1� x)u (3.23)_� = �1 + (a+ u)� (3.24)

66 Chapter 3. Variable-structure systemswhich we may also write in vector formddt 24 x� 35 = 24 �axa�� 1 35+ 24 1� x� 35u: (3.25)In the phase plane we clearly have a switching curve given by the equationC(x; �) = 0. In the region de�ned by (1� x)� < c we have the dynamics_x = �ax; _� = a�� 1 (3.26)whereas in the region (1� x)� > c we have_x = �(a+ 1)x+ 1; _� = (a+ 1)�� 1: (3.27)We have to determine the relation of the two vector �elds that are de�nedin this way to the switching curve. For this purpose we compute the timederivative of the function y(t) := C(x(t); �(t)) along the trajectories of each ofthe two dynamics given by (3.26) and (3.27). Actually it turns out that theresult is the same in both cases and in fact doesn't even depend on u, sincethe gradient of C which is given by[@C@x @C@� ] = [�� 1� x] (3.28)is orthogonal to the input vector �eld in (3.25). We get_y = a�+ x� 1: (3.29)So the mode selected at a given point on the switching curve is determined bythe sign of the quantity a�+x� 1. If a�+x� 1 > 0, then we enter the region(1 � x)� > c corresponding to the mode u = 1. If a� + x � 1 < 0 then theselected mode is the one in which u = 0. Since both dynamics agree on thesign of _y the sliding mode cannot occur as long as a�+ x� 1 6= 0.We still have to consider the possibility of a sliding mode at the point on theswitching curve at which a�+x� 1 = 0, if such a point does exist. It is easilyveri�ed that there is indeed such a point in the region of interest (0 � x � 1)if ac � 1. Because we already know that a sliding mode cannot exist at otherpoints in the (x; �)-plane, a sliding regime must create an equilibrium. Sinceat the intersection point of the curves y = 0 and _y = 0 the gradient of Cannihilates not only the input vector in (3.25) but also the drift vector, thesetwo vectors are linearly dependent. This means that the two tangent vectorscorresponding to u = 0 and u = 1 are also linearly dependent. If these vectorspoint in opposite directions, we can indeed create a sliding mode by choosingu 2 [0; 1] such that the right hand side in (3.25) vanishes. The three equationsin three unknowns given by�ax+ (1� x)u = 0a�� 1 + �u = 0�(1� x) = c

3.6. Notes and references for Chapter 3 67are solved in the region 0 � x � 1 byx = 1�pac; � =pc=a; u =pa=c� a: (3.30)So the sliding mode exists when 0 �pa=c� a � 1, that is, whena(a+ 1)2 � c � 1a: (3.31)Assume now that this condition holds, and consider the dynamics (3.21{3.25)with the initial conditionx0 = 1�pac; �0 =pc=a: (3.32)By computing �y it can be veri�ed that from this initial conditions there arethree possible continuations, corresponding to the three choices u = 0, u = 1and u =pa=c�a. This actually means that there are in�nitely many solutionsthat have initial condition (3.32), since one can take u(t) =pa=c�a for somearbitrary positive time and then continue with either u = 0 or u = 1. Some ofthe trajectories of the system (3.21{3.25) for the parameter values a = 12 andc = 1 are illustrated in Fig. 3.1.We conclude that the system (3.21{3.25) is not well-posed as an initial-value problem if the parameters a and c satisfy (3.31). However it should benoted that the necessary conditions for the original optimization problem donot have the form of an initial-value problem but rather of a mixed boundaryvalue problem, since we have the mixed boundary conditions x(0) = x0 and�(T ) = 0. As such the problem is well-posed and the option of \lying still" atthe equilibrium point created by the sliding mode is crucial. Also, the meaningof this equilibrium point and the corresponding value of u is clear: apart from\initial" and \�nal" e�ects, this value of u is the one that indicates optimaladvertizing expenditures. Note that it would be easy to rewrite the suggestedoptimal policy as a feedback policy, with a feedback mapping that dependsdiscontinuously on x and t, and that assumes only three di�erent values.3.6 Notes and references for Chapter 3This chapter obviously leans heavily on the work by Filippov as described inhis book [54]. The book, which appeared �rst in Russian in 1985, �ts intoa tradition of research in non-smooth dynamical systems that spans severaldecades. The solution concept proposed by Filippov originally in 1960 [53]uses the theory of di�erential inclusions, which is already in itself an importantbranch of mathematical analysis; see for instance [12]. In the present book ourpoint of view is somewhat di�erent. Rather than trying to infer in some waythe behavior of the sliding mode which takes place on a lower-dimensionalmanifold from the system's behavior in the \main" modes which take placeon full-dimensional regions, we consider all modes in principle on the same

68 Chapter 3. Variable-structure systems

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

Figure 3.1: Trajectories for Vidale-Wolfe example. Horizontal: x, vertical: �.Dotted: switching curve, with u = 0 to the right and u = 1 to the left. Dashed:trajectories leading to and from the equilibrium point

3.6. Notes and references for Chapter 3 69footing. We believe that this \hybrid" perspective is natural in many casesand can also be used e�ectively for the type of systems considered by Filippovand other authors in the same tradition. The hybrid point of view will beworked out further in the following chapters.

70 Chapter 3. Variable-structure systems

Chapter 4Complementarity systemsWe have already seen several examples of situations in which modes are deter-mined by pairs of so-called \complementary variables". Two scalar variablesare said to be complementary if they are both subject to an inequality con-straint, and if at all times at least one of these constraints is satis�ed withequality. The most obvious example is that of the ideal diode. In this casethe complementary variables are the voltage across the diode and the currentthrough it. When the voltage drop across the diode is negative the currentmust be zero, and the diode is said to be in nonconducting mode; when the cur-rent is positive the voltage must be zero, and the diode is in conducting mode.There are many more examples of hybrid systems in which mode switching isdetermined by complementarity conditions. We call such systems complemen-tarity systems . As we shall see, complementarity conditions arise naturallyin a number of applications; moreover, in several other applications one mayrewrite a given system of equations and inequalities in complementarity formby a judicious choice of variables.As a matter of convention, we shall always normalize complementary vari-ables in such a way that both variables in the pair are constrained to be non-negative; note that this deviates from standard sign conventions for diodes. Soa pair of variables (u; y) is said to be subject to a complementarity conditionif the following holds:u � 0; y � 0; ������ y = 0u = 0: (4.1)Often we will be working with several pairs of complementary variables. Forsuch situations it is useful to have a vector notation available. We shall saythat two vectors of variables (of equal length) are complementary if for all ithe pair of variables (ui; yi) is subject to a complementarity condition. In themathematical programming literature, the notation0 � y ? u � 0 (4.2)is often used to indicate that two vectors are complementary. Note that theinequalities are taken in a componentwise sense, and that the usual interpre-tation of the \perp" symbol (namelyPi yiui = 0) does indeed, in conjunction71

72 Chapter 4. Complementarity systemswith the inequality constraints, lead to the condition fyi = 0g _ fui = 0gfor all i. Alternatively, one might say that the \perp" relation is also takencomponentwise.Therefore, complementarity systems are systems whose ow conditions canbe written in the formf( _x; x; y; u) = 0 (4.3a)0 � y ? u � 0: (4.3b)In this formulation, the variables yi and ui play completely symmetric roles.Often it is possible to choose the denotations yi and ui in such a way that theconditions actually appear in the convenient \semi-explicit" form_x = f(x; u) (4.4a)y = h(x; u) (4.4b)0 � y ? u � 0: (4.4c)The ow conditions (4.3) or (4.4) still have to be supplemented by appropriateevent conditions which describe what happens when there is a switch betweenmodes. In some applications it will be enough to work with the default eventconditions that require continuity across events; in other applications one needsmore elaborate conditions.Additional continuous input (or \control") variables may of course easilybe added to a system description such as (4.4); discrete input variables mightbe added as well. In this chapter, however, we shall mainly be concerned withclosed systems in which such additional inputs do not appear. The motivationfor doing this is that we need an understanding of the dynamics of closedsystems before we can discuss systems with inputs. It may be noted, though,that the dynamical system (4.3a) (or (4.4a{4.4b)) taken as such is an opensystem, which is \closed" by adding the complementarity conditions (4.4c).Therefore, the theory of open (or \input-output") dynamical systems will stillplay an important role in this chapter.In the mathematical programming literature, the so-called linear comple-mentarity problem (LCP) has received much attention; see the book [39] foran extensive survey. The LCP takes as data a real k-vector q and a real k� kmatrix M , and asks whether it is possible to �nd k-vectors u and y such thaty = q +Mu; 0 � y ? u � 0: (4.5)The main result on the linear complementarity problem that will be used belowis the following [135], [39, Thm. 3.3.7]: the LCP (4.5) is uniquely solvable forall data vectors q if and only if all principal minors of the matrix M arepositive. (Given a matrix M of size k � k and two nonempty subsets I andJ of f1; : : : ; kg of equal cardinality, the (I; J)-minor of M is the determinantof the square submatrix MIJ := (mij)i2I; j2J . The principal minors are thosewith I = J [55, p. 2].) A matrix all of whose minors are positive is said to bea P-matrix.

4.1. Examples 734.1 Examples4.1.1 Circuits with ideal diodesA large amount of electrical network modeling is carried out on the basis ofideal lumped elements: resistors, inductors, capacitors, diodes, and so on.There is not necessarily a one-to-one relation between the elements in a modeland the parts of the actual circuit; for instance, a resistor may under somecircumstances be better modeled by a parallel connection of an ideal resistorand an ideal capacitor than by an ideal resistor alone. The standard idealelements should rather be looked at as forming a construction kit from whichone can quickly build a variety of models.Among the standard elements the ideal diode has a special position becauseof the nonsmoothness of its characteristic. In circuit simulation software thathas no ability to cope with mode changes, the ideal diode cannot be admittedas a building block and will have to be replaced for instance by a heavilynonlinear resistor; a price will have to be paid in terms of speed of simulation.The alternative is to work with a hybrid system simulator; more speci�cally,the software will have to be able to deal with complementarity systems.To write the equations of a network with (say) k ideal diodes in complemen-tarity form, �rst extract the diodes so that the network appears as a k-port.For each port, we have a choice between denoting voltage by ui and current byyi or vice versa (with the appropriate sign conventions). Often it is possibleto make these choices in such a way that the dynamics of the k-port can bewritten as _x = f(x; u); y = h(x; u):For linear networks, one can actually show that it is always possible to writethe dynamics in this form. To achieve this, it may be necessary to let uidenote voltage at some ports and current at some other ports; in that caseone sometimes speaks of a \hybrid" representation, where of course the termis used in a di�erent sense than the one used in this book. Replacing the portsby diodes, we obtain a representation in the semi-explicit complementarityform (4.4).For electrical networks it is often reasonable to assume that there are nojumps in the continuous state variables, so that there is no need to specifyevent conditions in addition to the ow conditions (4.4). Complementaritysystems in general do not always have continuous solutions, so if one wantsto prove that electrical networks with ideal diodes do indeed have continuoussolutions, one will have to make a connection with certain speci�c propertiesof electrical networks. The passivity property is one that immediately comesto mind, and indeed there are certain conclusions that can be drawn frompassivity and that are relevant in the study of properies of complementaritysystems. To illustrate this, consider the speci�c case of a linear passive systemcoupled to a number of ideal diodes. The system is described by equations of

74 Chapter 4. Complementarity systemsthe form_x = Ax+Buy = Cx+Du0 � y ? u � 0: (4.6)Under the assumption that the system representation is minimal, the passivityproperty implies (see [155]) that there exists a positive de�nite matrix Q suchthat 24 ATQ+QA QB � CTBTQ� C �(D +DT ) 35 � 0: (4.7)If now for instance the matrix D is nonsingular, then it follows that D isactually positive de�nite. Under this condition one can prove that the com-plementarity system (4.6) has continuous solutions. If on the other hand D isequal to zero, then the passivity condition (4.7) implies that C = BTQ so thatin this case the matrix CB = BTQB is positive de�nite (assuming that B hasfull column rank). Under this condition the system (4.6) has solutions withcontinuous state trajectories, if the system is consistently initialized, i. e. theinitial condition x0 satis�es Cx0 � 0. See [34] for proofs and additional infor-mation on the nature of solutions to linear passive complementarity systems.The importance of the matrices D and CB is related to the fact that theyappear in the power series expansion of the transfer matrix C(sI�A)�1B+Daround in�nity:C(sI �A)�1B +D = D + CBs�1 + CABs�2 + � � � :We will return to this when we discuss linear complementarity systems.4.1.2 Mechanical systems with unilateral constraintsMechanical systems with geometric inequality constraints (i. e. inequality con-straints on the position variables, such as in the simple example of Figure 2.7)are given by equations of the following form (see [138]), in which @H@p and @H@qdenote column vectors of partial derivatives, and the time arguments of q, p,y, and u have been omitted for brevity:_q = @H@p (q; p) q 2 Rn ; p 2 Rn (4.8a)_p = �@H@q (q; p) + @CT@q (q)u; u 2 Rk (4.8b)y = C(q); y 2 Rk (4.8c)0 � y ? u � 0: (4.8d)

4.1. Examples 75Here, C(q) � 0 is the column vector of geometric inequality constraints, andu � 0 is the vector of Lagrange multipliers producing the constraint forcevector (@C=@q)T (q)u. (The expression @CT =@q denotes an n�k matrix whosei-th column is given by @Ci=@q.) The perpendicularity condition expressesin particular that the i-th component of ui can only be non-zero if the i-thconstraint is active, that is, yi = Ci(q) = 0. The appearance of the reactionforce in the above form, with ui � 0, can be derived from the principle thatthe reaction forces do not exert any work along virtual displacements that arecompatible with the constraints. This basic principle of handling geometricinequality constraints can be found e. g. in [125, 86], and dates back to Fourierand Farkas.The Hamiltonian H(q; p) denotes the total energy, generally given as thesum of a kinetic energy 12pTM�1(q)p (where M(q) denotes the mass matrix,depending on the con�guration vector q) and a potential energy V (q). Thesemi-explicit complementarity system (4.8) is called a Hamiltonian comple-mentarity system, since the dynamics of every mode is Hamiltonian [138]. Inparticular, every mode is energy-conserving, since the constraint forces areworkless. It should be noted though that the model could be easily extendedto mechanical systems with dissipation by replacing the second set of equationsof (4.8) by_p = �@H@q (q; p)� @R@ _q ( _q) + @CT@q (q)u (4.9)where R( _q) denotes a Rayleigh dissipation function.4.1.3 Optimal control with state constraintsThe purpose of this subsection is to indicate in which way one may relate opti-mal control problems with state constraints to complementarity systems. Thestudy of this subject is far from being complete; we will o�er some suggestionsrather than present a rigorous treatment. Consider the problem of maximizinga functional of the formZ T0 F (t; x(t); u(t))dt + FT (x(T )) (4.10)over a collection of trajectories described by_x(t) = f(t; x(t); u(t)); x(0) = x0 (4.11)together with the constraintsg(t; x(t); u(t)) � 0: (4.12)In the above, g may be a vector-valued function, and then the inequalitiesare taken componentwise. Under suitable conditions (see [65] for much more

76 Chapter 4. Complementarity systemsdetailed information), candidates for optimal solutions can be found by solvinga system of equations that is obtained as follows. Let � be a vector variableof the same length as x, and de�ne the Hamiltonian H(t; x; u; �) byH(t; x; u; �) = F (t; x; u) + �T f(t; x; u): (4.13)Also, let � be a vector of the same length as g, and de�ne the LagrangianL(t; x; u; �; �) byL(t; x; u; �; �) = H(t; x; u; �) + �T g(t; x; u): (4.14)The system referred to before is now the following:_x(t) = f(t; x(t); u(t)) (4.15a)_�(t) = �@L@x (t; x(t); u(t); �(t); �(t)) (4.15b)u(t) = arg maxfujg(t;x(t);u)�0gL(t; x(t); u; �(t); �(t)) (4.15c)0 � g(t; x(t); u(t)) ? �(t) � 0 (4.15d)with initial conditionsx(0) = x0 (4.16)and �nal conditions�(T ) = @FT@x (x(T )): (4.17)Suppose that u(t) can be solved from (4.15c) so thatu(t) = u�(t; x(t); �(t); �(t)) (4.18)where u�(t; x; �; �) is a certain function. Then de�ne g�(t; x; �; �) byg�(t; x; �; �) = g(t; x; u�(t; x; �; �)) (4.19)and introduce a variable y(t) byy(t) = g�(t; x(t); �(t); �(t)): (4.20)The system (4.15) can now be rewritten as_x(t) = f(t; x(t); u�(t; x(t); �(t); �(t)))_�(t) = �@L@x (t; x(t); u�(t; x(t); �(t); �(t)); �(t); �(t))y(t) = g�(t; x(t); �(t); �(t))0 � y(t) ? �(t) � 0: (4.21)

4.1. Examples 77Here we have a (time-inhomogeneous) complementarity system with state vari-ables x and � and complementary variables y and �. The system has mixedboundary conditions (4.16{4.17); therefore one will have existence and unique-ness of solutions under conditions that in general will be di�erent from the onesthat hold for initial-value problems.A case of special interest is the one in which a quadratic criterion is op-timized for a linear time-invariant system, subject to linear inequality con-straints on the state. Consider for instance the following problem: minimize12 Z T0 (x(t)TQx(t) + u(t)Tu(t))dt (4.22)subject to_x(t) = Ax(t) +Bu(t); x(0) = x0 (4.23)Cx(t) � 0 (4.24)where A, B, and C are matrices of appropriate sizes, and Q is a nonnegativede�nite matrix. Following the scheme above leads to the system_x = Ax+Bu; x(0) = x0 (4.25a)_� = Qx�AT�� CT �; �(T ) = 0 (4.25b)u = arg max[� 12uTu+ �TBu] (4.25c)0 � Cx ? � � 0 (4.25d)where we have omitted the time arguments for brevity. Solving for u from(4.25c) leads to the equationsddt 24 x� 35 = 24 A BBTQ �AT 3524 x� 35+ 24 0�CT 35 � (4.26a)y = [C 0]24 x� 35 (4.26b)0 � y ? � � 0: (4.26c)Not surprisingly, this is a linear Hamiltonian complementarity system.The study of optimal control problems subject to state constraints isfraught with di�culties; see Hartl et al. [65] for a discussion. The settingof complementarity systems may be of help in overcoming part of these di�-culties.4.1.4 Variable-structure systemsConsider a nonlinear input-output system of the form_x = f(x; �u); �y = h(x; �u) (4.27)

78 Chapter 4. Complementarity systemsin which the input and output variables are adorned with a bar for reasonsthat will become clear in a moment. Suppose that the system is in feedbackcoupling with a relay element given by��������� �u = 1; �y � 0�1 � �u � 1; �y = 0�u = �1; �y � 0: (4.28)As we have seen above, many of the systems considered in the well-known bookby Filippov on discontinuous dynamical systems [54] can be rewritten in thisform. At �rst sight, relay systems do not seem to �t in the complementarityframework. However, let us introduce new variables y1, y2, u1, and u2, togetherwith the following new equations:u1 = 12 (1� �u)u2 = 12 (1 + �u) (4.29)�y = y1 � y2Instead of considering (4.27) together with (4.28), we can also consider (4.27)together with the standard complementarity conditions for the vectors y =col(y1; y2) and u = col(u1; u2):������ y1 = 0; u1 � 0y1 � 0; u1 = 0 ; ������ y2 = 0; u2 � 0y2 � 0; u2 = 0: (4.30)It can be veri�ed easily that the trajectories of (4.27{4.29{4.30) are the sameas those of (4.27{4.28). Note in particular that, although (4.30) in principleallows four modes, the conditions (4.29) imply that u1 + u2 = 1 so that themode in which both u1 and u2 vanish is excluded, and the actual number ofmodes is three.So it turns out that we can rewrite a relay system as a complementaritysystem, at least if we are willing to accept that some algebraic equations appearin the system description. It is possible to eliminate the variables �y and �u andobtain equations in the form_x = f(x; u2 � u1)y1 � y2 = h(x; u2 � u1)u1 + u2 = 1 (4.31)together with the complementarity conditions (4.30), but (4.31) is not in stan-dard input-state-output form but rather in a DAE type of formF ( _x; x; y; u) = 0: (4.32)

4.1. Examples 79If the relay is a part of a model whose equations are built up from submodelsthen it is likely anyway that the system description will already be in termsof both di�erential and algebraic equations, and then it may not be much ofa problem to have a few algebraic equations added (depending on how the\index" [28] of the system is a�ected). Alternatively however one may replacethe equations (4.29) byu1 = 12 (1� �u)y2 = 12 (1 + �u) (4.33)�y = y1 � u2which are the same as (4.29) except that y2 and u2 have traded places. Theequations (4.31) can now be rewritten as_x = f(x; 1� 2u1)y1 = h(x; 1� 2u1) + u2y2 = 1� u1 (4.34)and this system does appear in standard input-output form. The only conces-sion one has to make here is that (4.34) will have a feedthrough term (i. e. theoutput y depends directly on the input u) even when this is not the case inthe original system (4.27).4.1.5 A class of piecewise linear systemsSuppose that a linear system is coupled to a control device which switchesbetween several linear low-level controllers depending on the state of the con-trolled system, as is the case for instance in many gain scheduling controllers;then the closed-loop system may be described as a piecewise linear system.Another way in which piecewise linear systems may arise is as approximationsto nonlinear systems. Modeling by means of piecewise linear systems is at-tractive because it combines the relative tractability of linear dynamics witha exibility that is often needed for a precise description of dynamics over arange of operating conditions.There exist de�nitions of piecewise linear systems at various levels of gen-erality. Here we shall limit ourselves to systems of the following form (timearguments omitted for brevity):_x = Ax+Bu (4.35a)y = Cx+Du (4.35b)(yi; ui) 2 graph(fi) (i = 1; : : : ; k) (4.35c)where, for each i, fi is a piecewise linear function from R to R2 . As is commonusage, we use the term \piecewise linear" to refer to functions that would infact be more accurately described as being piecewise a�ne. We shall consider

80 Chapter 4. Complementarity systemsfunctions fi that are continuous, although from some points of view it wouldbe natural to include also discontinuous functions; for instance systems inwhich the dynamics is described by means of piecewise constant functionshave attracted attention in hybrid systems theory.The model (4.35) is natural for instance as a description of electrical net-works with a number of piecewise linear resistors. Descriptions of this formare quite common in circuit theory (cf. [93]). Linear relay systems are alsocovered by (4.35); note that the \sliding mode" corresponding to the verti-cal part of the relay characteristic is automatically included. Piecewise linearfriction models are often used in mechanics (for instance Coulomb friction),which again leads to models like (4.35); see Subsections 2.2.10 and 2.2.11.One needs to de�ne a solution concept for (4.35); in particular, one has tosay in what function space one will be looking for solutions. With an eye onthe intended applications, it seems reasonable to require that the trajectoriesof the variable x should be continuous and piecewise di�entiable. As for thevariable u, some applications suggest that it may also be too much to requirecontinuity for this variable. For example, take a mass point that is connectedby a linear spring to a �xed wall, and that can move in one direction subject toCoulomb friction. In a model for this situation the variable u would play therole of the friction force which, according to the Coulomb model, has constantmagnitude as long as the mass point is moving, and has sign opposite to thedirection of motion. If the mass point is given a su�ciently large initial velocityaway from the �xed wall, it will come to a standstill after some time and thenimmediately be pulled back towards the wall, so that in this case the frictionforce jumps instantaneously from one end of its interval of possible values tothe other. Even allowing jumps in the variable u, we can still de�ne a solutionof (4.35) to be a triple (x; u; y) such that (4.35b) and (4.35c) hold for almostall t, and (4.35a) is satis�ed in the sense of Carath�eodory, that is to sayx(t) = x(0) + Z t0 [Ax(�) +Bu(�)]d� (4.36)for all t.The �rst question that should be answered in connection with the system(4.35) is whether solutions exist and are unique. For this, one should �rst ofall �nd conditions under which, for a given initial condition x(0) = x0, thereexists a unique continuation in one of the possible \modes" of the systems(corresponding to all possible combinations of the di�erent branches of thepiecewise linear characteristics of the system). This can be a highly nontrivialproblem; for instance in a mechanical system with many friction points, it maynot be so easy to say at which points sliding will take place and at which pointsstick will occur. It turns out to be possible to address the problem on the basisof the theory of the linear complementarity problem and extensions of it. Forthe case of Coulomb friction, also in combination with nonlinear dynamics,this is worked out in [129]. The general case can be developed on the basisof a theorem by Kaneko and Pang [85], which states that any piecewise linear

4.1. Examples 81characteristic can be described by means of the so-called Extended HorizontalLinear Complementarity Problem. On this basis, the piecewise linear system(4.35) may also be described as an extended horizontal linear complementaritysystem. Results on the solvability of the EHLCP have been given by Sznajderand Gowda [148]. Using these results, one can obtain su�cient conditions forthe existence of unique solution starting at a given initial state; see [33] fordetails.4.1.6 Projected dynamical systemsThe concept of equilibrium is central to mathematical economics. For instance,one may consider an oligopolistic market in which several competitors deter-mine their production levels so as to maximize their pro�ts; it is of interest tostudy the equilibria that may exist in such a situation. On a wider scale, onemay discuss general economic equilibrium involving production, consumption,and prices of commodities. In fact in all kinds of competitive systems thenotion of equilibrium is important.The term \equilibrium" can actually be understood in several ways. Forinstance, the celebrated Nash equilibrium concept of game theory is de�ned asa situation in which no player can gain by unilaterally changing his position.Similar notions in mathematical economics lead to concepts of equilibria thatcan be characterized in terms of systems of algebraic equations and inequalities.On the other hand, we have the classical notion of equilibrium in the theoryof dynamical systems, where the concept is de�ned in terms of a given set ofdi�erential equations. It is natural to expect, though, that certain relationscan be found between the static and dynamic equilibrium concepts.In [49], Dupuis and Nagurney have proposed a general strategy for em-bedding a given static equilibrium problem into a dynamic system. Dupuisand Nagurney assume that the static equilibrium problem can be formulatedin terms of a variational equality ; that is to say, the problem is speci�ed bygiving a closed convex subset K of Rk and a function F from K to Rk , and�x 2 K is an equilibrium ifhF (�x); x� �xi � 0 (4.37)for all x 2 K. The formulation in such terms is standard within mathematicalprogramming. With the variational problem they associate a discontinuousdynamical system that is de�ned by _x = �F (x) on the interior of K but thatis de�ned di�erently on the boundary of K in such a way as to make sure thatsolutions will not leave the convex set K. They then prove that the stationarypoints of the so de�ned dynamical system coincide with the solutions of thevariational equality.In some more detail, the construction proposed by Dupuis and Nagurneycan be described as follows. The space Rk in which state vectors take theirvalues is taken as a Euclidean space with the usual inner product. Let P denotethe mapping that assigns to a given point x in Rk the (uniquely de�ned) point

82 Chapter 4. Complementarity systemsin K that is closest to x; that is to say,P (x) = arg minz2K jjx� zjj: (4.38)For x 2 K and a velocity vector v 2 Rk , let�(x; v) = lim�!0 P (x+ �v)� x� : (4.39)If x is in the interior of K, then clearly �(x; v) = v; however if x is on theboundary of K and v points outwards then �(x; v) is a modi�cation of v. Thedynamical system considered by Dupuis and Nagurney is now de�ned by_x = �(x;�F (x)) (4.40)with initial condition x0 inK. The right hand side of this equation is in generalsubject to a discontinuous change when the state vector reaches the boundaryof K. The state may then follow the boundary along a (k � 1)-dimensionalsurface or a part of the boundary characterized by more than one constraint,and after some time it may re-enter the interior of K after which it may againreach the boundary, and so on.In addition to the expression (4.39) Dupuis and Nagurney also employ adi�erent formulation which has been used in [48]. For this, �rst introduce theset of inward normals which is de�ned, for a boundary point x of K, byn(x) = f j jj jj = 1; and h ; x� yi � 0; 8y 2 Kg: (4.41)If K is a convex polyhedron then the vector de�ned in (4.39) may equivalentlybe described by�(x; v) = v + hv;� �i � (4.42)where � is de�ned by � := arg max 2n(x)hv;� i: (4.43)A further reformulation is possible by introducing the \cone of admissi-ble velocities". To formulate this concept, �rst recall that a curve in Rk is asmooth mapping from an interval, say (�1; 1), to Rk . An admissible velocityat a point x with respect to the closed convex set K � Rk is any vector thatappears as a directional derivative at 0 of a C1 function f(t) that satis�esf(0) = x and f(t) 2 K for t � 0. One can show that the set of admissiblevelocities is a closed convex cone for any x in the boundary of K; of course,the set of admissible velocities is empty when x 62 K and coincides with Rkif x belongs to the interior of K. One can furthermore show (see [71]) thatthe mapping de�ned in (4.42) for given x is in fact just the projection to thecone of admissible velocities. In this way we get an alternative de�nition ofprojected dynamical systems. The new formulation is more \intrinsic" in a

4.1. Examples 83di�erential-geometric sense than the original one which is based on the stan-dard coordinatization of k-dimensional Euclidean space. Indeed it would bepossible in this way to formulate projected dynamics for systems de�ned onRiemannian manifolds; the inner product on tangent spaces that is providedby the Riemannian structure makes it possible to de�ne the required projec-tion. One possible application would be the use of projected gradient ows to�nd minima subject to constraints; cf. for instance [75] for the unconstrainedcase.Assume now that the set K is given as an intersection of convex sets ofthe form fx j hi(x) � 0g where the functions hi are smooth. This is actuallythe situation that one typically �nds in applications. It is then possible toreformulate the projected dynamical system as a complementarity system.The construction is described in [71] and we summarize it brie y here. LetH(x) denote the gradient matrix de�ned by the functions hi(x); that is to say,the (i; j)-th element of H(x) is(H(x))ij = @hi@xj (x): (4.44)For x 2 K, let I(x) be the set of \active" indices, that is,I(x) = fi j hi(x) = 0g: (4.45)We denote by HI(x)� the matrix formed by the rows of H(x) whose indices areactive; it will be assumed that this matrix has full row rank for all x in theboundary of K (\independent constraints"). One can then show that for eachx 2 K the cone of admissible velocities is given by fv j HI(x)�v � 0g. Moreover,the set of inward normals as de�ned in (4.41) is given by f j jj jj = 1 and =HTI(x)�u for some u � 0g. Consequently, the projection of an arbitrary vectorv0 to the cone of admissible velocities is obtained by solving the minimizationproblemminv fjjv0 � vjj j HI(x)�v � 0g:By standard methods, one �nds that the minimizer is given by HTI(x)�u whereu is the (unique) solution of the complementarity problem0 � HI(x)�v0 +HI(x)�HTI(x)�u ? u � 0: (4.46)Now, compare the projected dynamical system (4.40) to the complementaritysystem de�ned by_x = �F (x) +HT (x)u (4.47a)y = h(x) (4.47b)0 � y ? u � 0 (4.47c)where h(x) is a vector de�ned in the obvious manner by (h(x))i = hi(x), andwhere the trajectories of all variables are required to be continuous. Suppose

84 Chapter 4. Complementarity systemsthat the system is initialized at t = 0 at a point x0 in K. For indices i suchthat hi(x0) > 0, the complementarity conditions imply that we must haveui(0) = 0. For indices that are active at x0 we have yi(0) = 0; to satisfythe inequality constraints also for positive t we need _yi(0) � 0. Moreover,it follows from the complementarity conditions and the continuity conditionsthat we must have ui(0) = 0 for indices i such that _yi(0) = 0, and, vice versa,_yi(0) = 0 for indices i such that ui(0) > 0. Since_yI(x0)(0) = HI(x0)�(�F (x0) +HTu(0))= HI(x0)�(�F (x0) +HTI(x0)�uI(x0)(0))the vector uI(x0)(0) must be a solution of the complementarity problem (4.46).It follows that HTu(0) is of the form appearing in (4.42). The reverse con-clusion follows as well, and moreover one can show that \local" equality ofsolutions as just shown implies \global" equality [71].4.1.7 Di�usion with a free boundaryIn this subsection we consider a situation in which a complementarity systemarises as an approximation to a partial di�erential equation with a free bound-ary. We shall take a speci�c example which arises in the theory of optionpricing. For this we �rst need to introduce some terminology. A European putoption is a contract that gives the holder the right, but not the obligation, tosell a certain asset to the counterparty in the contract for a speci�ed price (the\exercise price") at a speci�ed time in the future (\time of maturity"). Theunderlying asset can for instance be a certain amount of stocks, or a certainamount of foreign currency. For a concrete example, consider an investor whohas stocks that are worth 100 now and who would like to turn these stocksinto cash in one year's time. Of course it is hard to predict what the value ofthe stocks will be at that time; to make sure that the proceeds will be at least90, the investor may buy a put option with exercise price 90 that matures inone year. In this way the investor is sure that she can sell the stocks for atleast 90.Of course one has to pay a price to buy such protection, and it is the purposeof option theory to determine \reasonable" option prices. The modern theoryof option pricing started in the early seventies with the seminal work by Black,Scholes, and Merton. This theory is not based on the law of large numbers,but rather on the observation that the risk that goes with conferring an optioncontract is not as big as it would seem to be at �rst sight. By following anactive trading strategy in the underlying asset, the seller (\writer") of theoption will be able to reduce the risk. Under suitable model assumptions therisk can even be completely eliminated; that is to say, the cost of providingprotection becomes independent of the evolution of the value of the underlyingasset and hence can be predicted in advance. The \no-arbitrage" argumentthen states that this �xed cost must, by the force of competition, be the marketprice of the option. The model assumptions under which one can show that

4.1. Examples 85the risk of writing an option can be eliminated are too strong to be completelyrealistic; nevertheless, they provide a good guideline for devising strategiesthat at least are able to reduce risk substantially.One of the assumptions made in the original work of Black and Scholes [20]is that the price paths of the underlying asset may be described by a stochasticdi�erential equation of the formdS(t) = �S(t)dt+ �S(t)dw(t) (4.48)where w(t) denotes a standardWiener process. (See any textbook on SDEs, forinstance [122], for the meaning of the above.) Under a number of additionalassumptions (for instance: the underlying asset can be traded continuouslyand without transaction costs, and there is one �xed interest rate r whichholds both for borrowing and for lending), Black and Scholes derived a partialdi�erential equation that describes the price of the option at any time beforematurity as a function of two variables, namely time t and the price S of theunderlying asset. The Black-Scholes equation for the option price C(S; t) is(with omission of the arguments)@C@t + 12�2S2 @2C@S2 + rS @C@S � rC = 0 (4.49)with end condition for time of maturity T and exercise price KC(S; T ) = max(K � S; 0) (4.50)and boundary conditionsC(0; t) = e�r(T�t)K; limS!1C(S; t) = 0: (4.51)It turns out that the \drift" parameter � in the equation (4.48) is immaterial,whereas the \volatility" parameter � is very important since it determines thedi�usion coe�cient in the PDE (4.49).So far we have been discussing a European put option. An American putoption is the same except that the option may be exercised at any time untilthe maturity date, rather than only at the time of maturity. (The terms\European" and \American" just serve as a way of distinction; both types ofoptions are traded in both continents.) The possibility of early exercise bringsa discrete element into the discussion, since at any time the option may be intwo states: \alive" or \exercised". For American options, the Black-Scholesequation (4.49) is replaced by an inequality@C@t + 12�2S2 @2C@S2 + rS @C@S � rC � 0 (4.52)in which equality holds if the option is not exercised, that is, if its value exceedsthe revenues of exercise. For the put option, this is expressed by the inequalityC(S; t) > max(K � S; 0): (4.53)

86 Chapter 4. Complementarity systemsThe Black-Scholes equation (4.49) is a nonlinear partial di�erential equation.A simple substitution, however, will transform it to a linear PDE. To this end,express the price of the underlying asset S in terms of a new (dimensionless)independent variable x byS = Kex:To make the option price dimensionless as well, introduce v := C=K. Wealso change the �nal value problem (4.49{4.50) to an initial value problem bysetting set � = T � t. After some computation, we �nd that the equation(4.49) is replaced by�@v@� + 12�2 @2v@x2 + (r � 12�2)@v@x � rv = 0: (4.54)with the inital and boundary conditions (for the European put option)v(x; 0) = max(1� ex; 0); limx!�1 v(x; �) = e�r� ; limx!1 v(x; �) = 0:(4.55)For the American put option, we get the set of inequalities@v@� � 12�2 @2v@x2 � (r � 12�2)@v@x + rv � 0 (4.56)v � max(1� ex; 0) (4.57)with the boundary conditionslimx!1 v(x; �) = 0 (4.58)and v(x; �) = 1� ex for x � xf (�) (4.59)where xf (�) is the location at time � of the free boundary which should bedetermined as part of the problem on the basis of the so-called \smooth past-ing" or \high contact" conditions which require that v and @v=@x should becontinuous as functions of x across the free boundary.De�ne the function g(x) byg(x) = max(1� ex; 0): (4.60)The partial di�erential inequality (4.56) and its associated boundary condi-tions may then be written in the following form which is implicit with respectto the free boundary:(@v@� � 12�2 @2v@x2 � (r � 12�2)@v@x + rv)(v � g) = 0 (4.61a)@v@� � 12�2 @2v@x2 � (r � 12�2)@v@x + rv � 0; v � g � 0: (4.61b)

4.1. Examples 87This already suggests a complementarity formulation. Indeed, the above mightbe considered as an example of an in�nite-dimensional complementarity sys-tem, both in the sense that the state space dimension is in�nite and in thesense that there are in�nitely many complementarity relations to be satis�ed.A complementarity system of the type that we usually study in this chap-ter is obtained by approximating the in�nite-dimensional system by a �nite-dimensional system. For the current application it seems reasonable to carryout the approximation by what is known in numerical analysis as the \methodof lines". In this approach there is a �rst stage in which the space variableis discretized but the time variable is not. Speci�cally, take a grid of, say, Npoints in the space variable (in our case this is the dimensionless price variablex), and associate to each grid point xi a variable xi(t) which is intended to bean approximation to v(xi; t). The action of the linear di�erential operatorv 7! 12�2 @2v@x2 + (r � 12�2)@v@x � rv (4.62)can be approximated by a linear mapping acting on the space spanned by thevariables x1; : : : ; xN . For instance, on an evenly spaced grid an approximationto the �rst-order di�erential operator @=@x is given byA1 = 12h

266666666666664�2 2 0 � � � � � � 0�1 0 1 0 ...0 �1 0 1 . . . ...... . . . . . . . . . . . . 0... 0 �1 0 10 � � � � � � 0 �2 2

377777777777775 (4.63)where h is the mesh size, and the second-order di�erential operator @2=@x2 isapproximated by

A2 = 1h2266666666666664�2 1 0 � � � � � � 01 �2 1 0 ...0 1 �2 1 . . . ...... . . . . . . . . . . . . 0... 0 1 �2 10 � � � � � � 0 1 �2

377777777777775 (4.64)The mapping (4.62) is then approximated by the matrixA = 12�2A2 + (r � 12�2)A1 � rI: (4.65)

88 Chapter 4. Complementarity systemsThe function g(x) is represented by the vector g with entries gi = g(xi).Consider now the linear complementary system with N+1 state variables andN pairs of complementary variables given by_x = 24 A 00 0 35 x+ 24 I0 35u (4.66a)y = [ I �g ]x (4.66b)0 � y ? u � 0: (4.66c)The system is initialized atxi(0) = gi (i = 1; : : : ; N); xN+1(0) = 1: (4.67)The complementarity system de�ned in this way is a natural candidate for pro-viding an approximate solution to the di�usion equation with a free boundarythat corresponds to the Black-Scholes equation for an American put option.A natural idea for generating approximate solutions of complementaritysystems is to use an implicit Euler method. For linear complementarity sys-tems of the form_x = Ax+Buy = Cx+Du0 � y ? u � 0 (4.68)the method may be written as follows:xk+1 � xk�t = Axk+1 +Buk+1 (4.69a)yk+1 = Cxk+1 +Duk+1 (4.69b)0 � yk+1 ? uk+1 � 0 (4.69c)where xk is intended to be an approximation to x(k�t), and similarly fory and u. At each step this gives rise to a complementarity problem whichunder suitable assumptions has a unique solution. The results of applyingthis method above to the equations for an American put option are shown inFig. 4.1; the �gure shows solutions for various times before expiry as a functionof the value of the underlying asset (the variable S).A more standard numerical method for dealing with the American-styleBlack-Scholes equation is the �nite-di�erence method in which both the timeand space variables are discretized; see for instance [158]. In general, an advan-tage of a \semidiscretization" approach over an approach in which all variablesare discretized simultaneously is that one may make use of the highly devel-oped theory of step size control for numerical solutions of ODEs, rather than

4.1. Examples 89

0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 4.1: Value of an American put option at di�erent times before expiry.using a uniform grid. We are of course working here with a complementar-ity system rather than an ODE, and it must be said that the theory of stepsize control for complementarity systems is at an early stage of development.Moreover, the theory of approximation of free-boundary problems by comple-mentarity systems has been presented here only for a special case and on thebasis of plausibility rather than formal proof, and much further work on thistopic is needed.4.1.8 Max-plus systemsFrom the fact that the relationz = max(x; y) (4.70)may also be written in the formz = x+ a = y + b; 0 � a ? b � 0 (4.71)it follows that any system that can be written in terms of linear operationsand the \max" operation can also be written as a complementarity system.In particular it follows that the so-called max-plus systems (see [13]), whichare closely related to timed Petri nets, can be written as complementaritysystems. The resulting equations appear in discrete time, as opposed to theother examples in this section which are all in continuous time; note however

90 Chapter 4. Complementarity systemsthat the \time" parameter in a max-plus system is in the standard applicationsa cycle counter rather than actual time. For further discussion of the relationbetween the max algebra and the complementarity problem see [44].4.2 Existence and uniqueness of solutionsHybrid systems provide a rather wide modeling context, so that there areno easily veri�able necessary and su�cient conditions for well-posedness ofgeneral hybrid dynamical systems. It is already of interest to give su�cientconditions for well-posedness of particular classes of hybrid systems, such ascomplementarity systems. The advantage of considering special classes is thatone can hope for conditions that are relatively easy to verify. In a number ofspecial cases, such as mechanical systems or electrical network models, thereare moreover natural candidates for such su�cient conditions.Uniqueness of solutions will below always be understood in the sense ofwhat is sometimes called right uniqueness , that is, uniqueness of solutionsde�ned on an interval [t0; t1) given an initial state at t0. It can easily happen ingeneral hybrid systems, and even in complementarity systems, that uniquenessholds in one direction of time but not in the other; take for instance the two-carts example of Subsection 2.2.9 with zero restitution coe�cient. We havehere one of the points in which discontinuous dynamical systems di�er fromsmooth systems. To allow for the possibility of an initial jump, one may letthe initial condition be given at t�0 .We have to distinguish between local and global existence and uniqueness.Local existence and uniqueness, for solutions starting at t0, holds if there existsan " > 0 such that on [t0; t0+") there is a unique solution starting at the giveninitial condition. For global existence and uniqueness, we require that forgiven initial condition there is a unique solution on [t0;1). If local uniquenessholds for all initial conditions and existence holds globally, then uniquenessmust also hold globally since there is no point at which solutions can split.However local existence does not imply global existence. This phenomenon isalready well-known in the theory of smooth dynamical systems; for instancethe di�erential equation _x(t) = x2(t) with x(0) = x0 has the unique solutionx(t) = x0(1 � x0t)�1 which for positive x0 is de�ned only on the interval[0; x�10 ). Some growth conditions have to be imposed to prevent this \escape toin�nity". In hybrid systems, there are additional reasons why global existencemay fail; in particular we may have an accumulation of mode switches. Anexample where such an accumulation occurs was already discussed in Chapter1 (see (1.12)). Another example comes from the so-called Fuller phenomenonin optimal control theory [106]. For the problem given by the equations _x1(t) =x2(t), _x2(t) = u(t) with end constraint x(T ) = 0, control constraints ju(t)j � 1,and cost functional R T0 x21(t)dt, it turns out that the optimal control is bang-bang (i. .e. it takes on only the two extreme values 1 and �1) and has anin�nite number of switches. As shown in [90], the phenomenon is not at allrare.

4.3. The mode selection problem 914.3 The mode selection problemAs noted in [138] it is not di�cult to �nd examples of complementarity systemsthat exhibit nonuniqueness of smooth continuations. For a simple example ofthis phenomenon within a switching control framework, consider the plant_x1 = x2; y = x2_x2 = �x1 � u (4.72)in closed-loop with a switching control scheme of relay typeu(t) = 1; if y(t) > 0�1 � u(t) � 1; if y(t) = 0u(t) = �1; if y(t) < 0: (4.73)(This could be interpreted as a mass-spring system subject to a \reversed"| and therefore non-physical | Coulomb friction.) It was shown in Sub-section 4.1.4 that such a variable-structure system can be modeled as a com-plementarity system. Note that from any initial (continuous) state x(0) =(x1(0); x2(0)) = (c; 0), with jcj � 1, there are three possible smooth continua-tions for t � 0 that are allowed by the equations and inequalities above:(i) x1(t) = x1(0); x2(t) = 0; u(t) = �x1(0); �1 � u(t) � 1;y(t) = x2(t) = 0(ii) x1(t) = �1 + (x1(0) + 1) cos t; x2(t) = �(x1(0) + 1) sin t;u(t) = 1; y(t) = x2(t) < 0(iii) x1(t) = 1 + (x1(0)� 1) cos t; x2(t) = �(x1(0)� 1) sin t;u(t) = �1; y(t) = x2(t) > 0:So the above closed-loop system is not well-posed as a dynamical system. Ifthe sign of the feedback coupling is reversed, however, there is only one smoothcontinuation from each initial state. This shows that well-posedness is a non-trivial issue to decide upon in a hybrid system, and in particular is a meaningfulperformance characteristic for hybrid systems arising from switching controlschemes.In this section we follow the treatment of [140] and consider systems of theform _x(t) = f(x(t); u(t)); x 2 Rn ; u 2 Rk (4.74a)y(t) = h(x(t); u(t)); y 2 Rk (4.74b)with the additional complementarity conditions0 � y(t) ? u(t) � 0: (4.75)

92 Chapter 4. Complementarity systemsThe functions f and h will always be assumed to be smooth.The complementarity conditions (4.75) imply that for some index set I �f1; : : : ; kg one has the algebraic constraintsyi(t) = 0 (i 2 I); ui(t) = 0 (i 62 I): (4.76)Note that (4.76) always represents k constraints which are to be taken inconjunction with the system of n di�erential equations in n + k variablesappearing in (4.74). The problem of determining which index set I has theproperty that the solution of (4.74{4.76) coincides with that of (4.74{4.75),at least on an initial time interval, is called the mode selection problem. Theindex set I represents the mode of the system.One approach to solving the mode selection problem would simply be totry all possibilities: solve (4.74) together with (4.76) for some chosen candidateindex set I , and see whether the computed solution is such that the inequalityconstraints y(t) � 0 and u(t) � 0 are satis�ed on some interval [0; "]. Un-der the assumption that smooth continuation is possible from x0, there mustat least be one index set for which the constraints will indeed be satis�ed.This method requires in the worst case the integration of 2k systems of n+ kdi�erential/algebraic equations in n+ k unknowns.In order to develop an alternative approach which leads to an algebraicproblem formulation, let us note �rst that we can derive from (4.74) a numberof relations between the successive time derivatives of y(�), evaluated at t = 0,and the same quantities derived from u(�). By successively di�erentiating(4.74b) and using (4.74a), we gety(t) = h(x(t); u(t));_y(t) = @h@x (x(t); u(t))f(x(t); u(t)) + @h@u(x(t); u(t)) _u(t)=: F1(x(t); u(t); _u(t));and in generaly(j)(t) = Fj(x(t); u(t); : : : ; u(j)(t)) (4.77)where Fj is a function that can be speci�ed explicitly in terms of f and h.From the complementarity conditions (4.75), it follows moreover that for eachindex i either(yi(0); _yi(0); : : : ) = 0 and (ui(0); _ui(0); : : : ) � 0 (4.78)or (yi(0); _yi(0); : : : ) � 0 and (ui(0); _ui(0); : : : ) = 0 (4.79)(or both), where we use the symbol � to denote lexicographic nonnegativity.(A sequence (a0; a1; : : : ) of real numbers is said to be lexicographically non-negative if either all ai are zero or the �rst nonzero element is positive.) This

4.3. The mode selection problem 93suggests the formulation of the following \dynamic complementarity problem."Problem DCP. Given smooth functions Fj : Rn+(j+1)k ! Rk (j = 0; 1; : : : ;)that are constructed from smooth functions f : Rn ! Rn and h : Rn ! Rkvia (4.77), �nd, for given x0 2 Rn , sequences (y0; y1; : : : ) and (u0; u1; : : : ) ofk-vectors such that for all j we haveyj = Fj(x0; u0; : : : ; uj) (4.80)and for each index i 2 f1; : : : ; kg at least one of the following is true:(y0i ; y1i ; : : : ) = 0 and (u0i ; u1i ; : : : ) � 0 (4.81)(y0i ; y1i ; : : : ) � 0 and (u0i ; u1i ; : : : ) = 0: (4.82)We shall also consider truncated versions where j only takes on the valuesfrom 0 up to some integer `; the corresponding problem will be denotedby DCP(`). It follows from the triangular structure of the equations thatif ((y0; : : : ; y`); (u0; : : : ; u`)) is a solution of DCP(`), then, for any `0 < `,((y0; : : : ; y`0); (u0; : : : ; u`0)) is a solution of DCP(`0). We call this the nestingproperty of solutions. We de�ne the active index set at stage `, denoted by I`,as the set of indices i for which (u0i ; : : : ; ui) � 0 in all solutions of DCP(`), sothat necessarily yji = 0 for all j in any solution of DCP (if one exists). Likewisewe de�ne the inactive index set at stage `, denoted by J`, as the set of indices ifor which (y0i ; : : : ; yi ) � 0 in all solutions of DCP(`), so that necessarily uji = 0for all j in any solution of DCP. Finally we de�ne K` as the complementaryindex set f1; : : : ; kgn(I`[J`). It follows from the nesting property of solutionsthat the index sets I` and J` are nondecreasing as functions of `. Since bothsequences are obviously bounded above, there must exist an index `� such thatI` = I`� and J` = J`� for all ` � `�. We �nally note that all index sets de�nedhere of course depend on x0; we suppress this dependence however to alleviatethe notation.The problem DCP is a generalization of the nonlinear complementarityproblem (NLCP) (see for instance [39]), which can be formulated as follows:given a smooth function F : Rk ! Rk , �nd k-vectors y and u such thaty = F (x; u) and 0 � y ? u � 0. For this reason the term \dynamic comple-mentarity problem" as used above seems natural. Apologies are due howeverto Chen and Mandelbaum who have used the same term in [37] to denote adi�erent although related problem.Computational methods for the NLCP form a highly active research subject(see [64] for a survey), due to the many applications in particular in equilib-rium programming. The DCP is a generalized and parametrized form of theNLCP and given the fact that the latter problem is already considered a majorcomputational challenge, one may wonder whether the approach taken in theprevious paragraphs can be viewed as promising from a computational pointof view. Fortunately, it turns out that under fairly mild assumptions the DCP

94 Chapter 4. Complementarity systemscan be reduced to a series of linear complementarity problems. In the contextof mechanical systems this idea is due to L�otstedt [96].To get a reduction to a sequence of LCPs, assume that the dynamics (4.74)can be written in the a�ne form_x(t) = f(x(t)) +Pki=1 gi(x(t))ui(t)y(t) = h(x(t)): (4.83)Extensive information on systems of this type is given for instance in [121]. Inparticular we need the following terminology. The relative degree of the i-thoutput yi is the number of times one has to di�erentiate yi to get a resultthat depends explicitly on the inputs u. The system is said to have constantuniform relative degree at x0 if the relative degrees of all outputs are the sameand are constant in a neighborhood of x0.We can now state the following theorem, in which we use the notationDCP(`) to indicate explicitly the dependence of the dynamic complementarityproblem on the number ` of di�erentiation steps. For a proof see [140]. Recallthat Lfh denotes the Lie derivative of h along the vector �eld given by f ; thatis, Lfh(x) = (@h=@x)(x)f(x). Also, the k-th Lie derivative Lkfh is de�nedinductively for k = 2; 3; : : : by Lkfh = Lf (Lk�1f h) with L1fh := Lfh.Theorem 4.3.1. Consider the system of equations (4.83) together with thecomplementarity conditions (4.75), and suppose that the system (4.83) hasconstant uniform relative degree � at a point x0 2 Rn . Suppose that x0 is suchthat (h(x0); : : : ; L��1f h(x0)) � 0 (4.84)(with componentwise interpretation of the lexicographic inequality), and suchthat all principal minors of the decoupling matrix LgL��1f h(x0) at x0 are posi-tive. For such x0, the dynamic complementarity problem DCP(`) has for each` a solution ((y0; : : : ; y`); (u0; : : : ; u`)) which can be found by solving a se-quence of LCPs. Moreover this solution is unique, except for the values of ujiwith i 62 J` and j > `� �.This result is algebraic in nature. We now return to di�erential equations;again the proof of the statement below is in [140].Theorem 4.3.2. Assume that the functions f , gi, and h appearing in (4.83)are analytic. Under the conditions of Thm. 4.3.1, there exists an " > 0such that (4.83-4.75) has a smooth solution with initial condition x0 on [0; "].Moreover, this solution is unique and corresponds to any mode I such thatI`� � I � I`� [K`� .

4.3. The mode selection problem 95Example 4.3.3. Consider the following nonlinear complementarity system:_x1 = 1� x2u (4.85a)_x2 = 1� x1u (4.85b)y = �x1x2 (4.85c)0 � y ? u � 0: (4.85d)The feasible set consists of the second and fourth quadrants of the plane. Inthe mode in which u = 0, the dynamics is obviously given by_x1 = 1; _x2 = 1: (4.86)In the other mode, it follows from _y = 0 that u must satisfy the equation(x21 + x22)u = x1 + x2 (4.87)which determines u as a function of x1 and x2 except in the origin. Thedynamics in the mode y = 0 is given for (x1; x2) 6= (0; 0) by_x1 = 1� x2x1 + x2x21 + x22 ; _x2 = 1� x1 x1 + x2x21 + x22 : (4.88)However, this can be simpli�ed considerably because when y = 0 we must havex1 = 0 or x2 = 0; in the �rst case we have_x1 = 0; _x2 = 1 (4.89)and in the second case_x1 = 1; _x2 = 0: (4.90)The system (4.85a{4.85c) has uniform relative degree 1 everywhere except atthe origin, where the relative degree is 3. The decoupling matrix is in this casejust a scalar and is given by x21 + x22. By Thm. 4.3.2, we �nd that everywhereexcept possibly at the origin a unique smooth continuation is possible. Thesituation at the origin needs to be considered separately. Thm. 4.3.2 doesnot apply, but we can use a direct argument. Whatever choice we make foru, if x1(0) = 0 and x2(0) = 0 then the equations (4.85a{4.85b) show that_x1(0) = 1 and _x2(0) = 1, so that any solution starting from the origin mustleave the feasible set. The same conclusion could be drawn from the DCP,since computation show that for trajectories starting from the origin we havey(0) = 0, _y(0) = 0, and �y(0) = �2 so that the DCP is not solvable. Weconclude that the system (4.85) as a whole is not well-posed.Example 4.3.4 (Passive systems). A system (4.83) is called passive (see[155]) if there exists a function V (x) � 0 (a storage function) such thatLfV (x) � 0LgiV (x) = hi(x); i = 1; � � � ; k: (4.91)

96 Chapter 4. Complementarity systemsLet us assume the following non-degeneracy condition on the storage functionV : rank �LgjLgiV (x)�i;j=1;��� ;k = k; for all x with h(x) � 0: (4.92)Since Lgjhi = LgjLgiV it follows that the system has uniform relative de-gree 1, with decoupling matrix D(x) given by the matrix in (4.92). If theprincipal minors of D(x) are all positive, then well-posedness follows. Notethat the condition of D(x) having positive principal minors corresponds to anadditional positivity condition on the storage function V . In fact, it can bechecked that for a linear system with quadratic storage function V (x) and nodirect feedthorugh from inputs to outputs, the decoupling matrix D(x) willbe positive de�nite if V (x) > 0 for x 6= 0. Hence, if the equations (4.83{4.75)represent a linear passive electrical network containing ideal diodes, then thissystem is well-posed.4.4 Linear complementarity systems4.4.1 Speci�cationConsider the following system of linear di�erential and algebraic equations andinequalities_x(t) = Ax(t) +Bu(t) (4.93a)y(t) = Cx(t) +Du(t) (4.93b)0 � y(t) ? u(t) � 0: (4.93c)The equations (4.93a) and (4.93b) constitute a linear system in state spaceform; the number of inputs is assumed to be equal to the number of outputs.The relations (4.93c) are the usual complementarity conditions . The set of in-dices for which yi(t) = 0 will be referred to as the active index set ; de�ning theactive index set in this way rather than by the condition ui(t) > 0 used in theprevious section allows us to write the inequality constraints corresponding toa given active index set as nonstrict inequalities rather than strict inequalities.The active index set is in general not constant in time, so that the systemswitches from one \operating mode" to another. To de�ne the dynamics of(4.93) completely, we will have to specify when these mode switches occur,what their e�ect will be on the state variables, and how a new mode will beselected. A proposal for answering these questions (cf. [69]) will be explainedbelow. The speci�cation of the complete dynamics of (4.93) de�nes a class ofdynamical systems called linear complementarity systems .Let n denote the length of the vector x(t) in the equations (4.93a{4.93b)and let k denote the number of inputs and outputs. There are then 2k possiblechoices for the active index set. The equations of motion when the active index

4.4. Linear complementarity systems 97set is I are given by_x(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t)yi(t) = 0; i 2 Iui(t) = 0; i 2 Ic (4.94)where Ic denotes the index set that is complementary to I , that is, Ic = fi 2f1; : : : ; kg j i 62 Ig. We shall say that the above equations represent the systemin mode I . An equivalent and somewhat more explicit form is given by the(generalized) state equations_x(t) = Ax(t) +B�IuI(t)0 = CI�x(t) +DIIuI(t) (4.95)together with the output equationsyIc(t) = CIc�x(t) +DIcIuI(t)uIc(t) = 0: (4.96)Here and below, the notation M�I , where M is a matrix of size m� k and Iis a subset of f1; : : : ; kg, denotes the submatrix of M formed by taking thecolumns ofM whose indices are in I . The notationMI� denotes the submatrixobtained by taking the rows with indices in the index set I .The system (4.95) will in general not have solutions in a classical sense forall possible initial conditions. The initial values of the variable x for whichthere does exist a continuously di�erentiable solution are called consistentstates . Under conditions that will be speci�ed below, each consistent initialstate gives rise to a unique solution of (4.95). The system (4.93) follows thepath of such a solution (it \stays in mode I") as long as the variables uI(t)de�ned implicitly by (4.95) and the variables yIc(t) de�ned by (4.96) are allnonnegative. As soon as continuation in mode I would lead to a violationof these inequality constraints, a switch to a di�erent mode has to occur. Incase the value of the variable x(t) at which violation of the constraints hasbecome imminent is not a consistent state for the new mode, a state jump iscalled for. So both concerning the dynamics in a given mode and concerningthe transitions between di�erent modes there are a number of questions to beanswered. For this we shall rely on the geometric theory of linear systems (see[160, 15, 89] for the general background).Denote by VI the consistent subspace of mode I , i. e. the set of initialconditions x0 for which there exist smooth functions x(�) and uI(�), with x(0) =x0, such that (4.95) is satis�ed. The space VI can be computed as the limit of

98 Chapter 4. Complementarity systemsthe sequence de�ned byV 0I = RnV i+1I = fx 2 V iI j 9u 2 RjIj s. t. Ax +B�Iu 2 V iI ; CI�x+DIIu = 0g:(4.97)There exists a linear mapping FI such that (4.95) will be satis�ed for x0 2 VIby taking uI(t) = FIx(t). The mapping FI is uniquely determined, and moregenerally the function uI(�) that satis�es (4.95) for given x0 2 VI is uniquelydetermined, if the full-column-rank conditionker�B�IDII � = f0g (4.98)holds and moreover we haveVI \ TI = f0g; (4.99)where TI is the subspace that can be computed as the limit of the followingsequence:T 0I = f0gT i+1I = fx 2 Rn j 9~x 2 T iI ; ~u 2 RjIj s. t.x = A~x +B�I ~u; CI�~x+DII ~u = 0g: (4.100)As will be indicated below, the subspace TI is best thought of as the jumpspace associated to mode I , that is, as the space along which fast motionswill occur that take an inconsistent initial state instantaneously to a point inthe consistent space VI ; note that under the condition (4.99) this projection isuniquely determined. The projection can be used to de�ne a jump rule. How-ever, there are 2k possible projections, corresponding to all possible subsetsof f1; : : : ; kg; which one of these to choose should be determined by a modeselection rule.For the formulation of a mode selection rule we have to relate in some wayindex sets to continuous states. Such a relation can be established on the basisof the so-called rational complementarity problem (RCP). The RCP is de�nedas follows. Let a rational vector q(s) of length k and a rational matrix M(s)of size k� k be given. The rational complementarity problem is to �nd a pairof rational vectors y(s) and u(s) (both of length k) such thaty(s) = q(s) +M(s)u(s) (4.101)and moreover for all indices 1 � i � k we have either yi(s) = 0 and ui(s) > 0for all su�ciently large s, or ui(s) = 0 and yi(s) > 0 for all su�ciently larges.1 The vector q(s) and the matrix M(s) are called the data of the RCP, and1Note the abuse of notation: we use q(s) both to denote the function s 7! q(s) and thevalue of that function at a speci�c point s 2 C . On a few occasions we shall also denoterational functions and rational vectors by single symbols without an argument, which is inprinciple the proper notation.

4.4. Linear complementarity systems 99we write RCP(q(s),M(s)). We shall also consider an RCP whose data are aquadruple of constant matrices (A;B;C;D) (such as could be used to de�ne(4.93a{4.93b)) and a constant vector x0, namely by settingq(s) = C(sI �A)�1x0 and M(s) = C(sI �A)�1B +D:We say that an index set I � f1; : : : ; kg solves the RCP (4.101) if there existsa solution (y(s); u(s)) with yi(s) = 0 for i 2 I and ui(s) = 0 for i 62 I . Thecollection of index sets I that solve RCP(A;B;C;D;x0) will be denoted byS(A;B;C;D;x0) or simply by S(x0) if the quadruple (A;B;C;D) is given bythe context.It is convenient to introduce an ordering relation on the �eld of rationalfunctions R(s). Given a rational function f(s), we shall say that f(s) is non-negative, and we write f � 0, if9�0 2 R 8� 2 R f� > �0 ) f(�) � 0g:An ordering relation between rational functions can now be de�ned by f � gif and only if f �g � 0. Note that this de�nes a total ordering on R(s) so thatwith this relation R(s) becomes an ordered �eld. Extending the conventionsthat we already have used for the real �eld we shall say that a rational vectoris nonnegative if and only if all its entries are nonnegative, and we shall writef(s) ? g(s), where f(s) and g(s) are rational vectors, if for each index i atleast one of the component functions fi(s) and gi(s) is identically zero. Withthese conventions, the rational complementarity problem may be written inthe formy(s) = q(s) +M(s)u(s)0 � y(s) ? u(s) � 0: (4.102)After these preparations, we can now proceed to a speci�cation of thecomplete dynamics of linear complementarity systems. We assume that aquadruple (A;B;C;D) is given whose transfer matrix G(s) = C(sI�A)�1B+D is totally invertible, i. e. for each index set I the k � k matrix GII (s) isnonsingular. Under this condition (see Thm. 4.4.1 below), the two subspacesVI and TI as de�ned above form for all I a direct sum decomposition of thestate space Rn , so that the projection along TI onto VI is well-de�ned. Wedenote this projection by PI . The interpretation that we give to the equations(4.93) is the following:������������������ _x = Ax+Bu; y = Cx+DuuI � 0; yI = 0; uIc = 0; yIc � 0I] 2 S(x); x] = PI]x: (4.103)Below we shall always consider the system (4.93) in the interpretation (4.103).

100 Chapter 4. Complementarity systems4.4.2 A distributional interpretationThe interpretation of TI as a jump space can be made precise by introducingthe class of impulsive-smooth distributions that was studied by Hautus [66](see also [67, 57]). The general form of an impulsive-smooth distribution � is� = p( ddt )� + f (4.104)where p(�) is a polynomial, ddt denotes the distributional derivative, � is thedelta distribution with support at zero, and f is a distribution that can beidenti�ed with the restriction to (0;1) of some function in C1(R). Theclass of such distributions will be denoted by Cimp. For an element of Cimpof the form (4.104), we write �(0+) for the limit value limt#0 f(t). Havingintroduced the class Cimp, we can replace the system of equations (4.95) byits distributional versionddtx = Ax+B�IuI + x0�0 = CI�x+DIIuI (4.105)in which the initial condition x0 appears explicitly, and we can look for asolution of (4.105) in the class of vector-valued impulsive-smooth distributions.It was shown in [67] that if the conditions (4.98) and (4.99) are satis�ed, thenthere exists a unique solution (x; uI) 2 Cn+jIjimp to (4.105) for each x0 2 VI+TI ;moreover, the solution is such that x(0+) is equal to PIx0, the projection ofx0 onto VI along TI . The solution is most easily written down in terms of itsLaplace transform:x(s) = (sI �A)�1x0 + (sI �A)�1B�I uI(s) (4.106)uI(s) = �G�1II (s)CI�(sI �A)�1x0; (4.107)whereGII(s) := CI�(sI �A)�1B�I +DII : (4.108)Note that the notation is consistent in the sense that GII(s) can also be viewedas the (I; I)-submatrix of the transfer matrix G(s) := C(sI � A)�1B +D. Itis shown in [67] (see also [117]) that the transfer matrix GII(s) associated tothe system parameters in (4.95) is left invertible when (4.98) and (4.99) aresatis�ed. Since the transfer matrices GII(s) that we consider are square, leftinvertibility is enough to imply invertibility, and so (by duality) we also haveVI + TI = Rn . Summarizing, we can list the following equivalent conditions.Theorem 4.4.1. Consider a time-invariant linear system with k inputs and koutputs, given by standard state space parameters (A;B;C;D). The followingconditions are equivalent.1. For each index set I � �k, the associated system (4.95) admits for eachx0 2 VI a unique smooth solution (x; u) such that x(0) = x0.

4.4. Linear complementarity systems 1012. For each index set I � �k, the associated distributional system (4.105)admits for each initial condition x0 a unique impulsive-smooth solution(x; u).3. The conditions (4.98) and (4.99) are satis�ed for all I � �k.4. The transfer matrix G(s) = C(sI � A)�1B +D is totally invertible (asa matrix over the �eld of rational functions).In connection with the system (4.93) it makes sense to introduce the followingde�nitions.De�nition 4.4.2. An impulsive-smooth distribution � = p( ddt)� + f as in(4.104) will be called initially nonnegative if the leading coe�cient of thepolynomial p(�) is positive, or, in case p = 0, the smooth function f is non-negative on some interval of the form (0; ") with " > 0. A vector-valuedimpulsive-smooth distribution will be called initially nonnegative if each of itscomponents is initially nonnegative in the above sense.De�nition 4.4.3. A triple of vector-valued impulsive-smooth distributions(u; x; y) will be called an initial solution to (4.93) with initial state x0 andsolution mode I if1. the triple (u; x; y) satis�es the distributional equationsddtx = Ax+Bu+ x0�y = Cx+Du2. both u and y are initially nonnegative3. yi = 0 for all i 2 I and ui = 0 for all i 62 I .For an impulsive-smooth distribution w that has a rational Laplace trans-form w(s) (such as in (4.106) and (4.107)), we have that w is initially nonneg-ative if and only if w(s) is nonnegative for all su�ciently large real values ofs. From this it follows that the collection of index sets I for which there existsan initial solution to (4.93) with initial state x0 and solution mode I is exactlyS(x0) as we de�ned this set before in terms of the rational complementarityproblem.An alternative approach to the construction of initial solutions for linearcomplementarity systems proceeds through the linear version of the dynamiccomplementarity problem that was discussed in Section 4.3. It has been shownby De Schutter and De Moor that the dynamic linear complementarity prob-lem can be rewritten as a version of the LCP known as the \extended linearcomplementarity problem" (see [45] for details).

102 Chapter 4. Complementarity systems4.4.3 Well-posednessMost of the well-posedness results that are available for linear complemen-tarity systems provide only su�cient conditions. For the case of bimodallinear complementarity systems, i. e. systems with only two modes (k = 1),well-posedness has been completely characterized however (see [73]; cf. also[138] for an earlier result with a slightly di�erent notion of well-posedness).Note that a system of the form (4.93a-4.93b) has a transfer function g(s) =C(sI �A)�1B +D which is a rational function. In this case the conditions ofThm. 4.4.1 apply if g(s) is nonzero. The Markov parameters of the system arethe coe�cients of the expansion of g(s) around in�nity,g(s) = g0 + g1s�1 + g2s�2 + � � � :The leading Markov parameter is the �rst parameter in the sequence g0, g1, : : :that is nonzero. In the theorem below it is assumed that the output matrix C isnonzero; note that if C = 0 and D 6= 0 the system (4.93) is just a complicatedway of representing the equations _x = Ax, so that in that case we do not reallyhave a bimodal system.Theorem 4.4.4. Consider the linear complimentarity system (4.93) underthe assumptions that k = 1 (only one pair of complementary variables) andC 6= 0; also assume that the transfer function g(s) = C(sI�A)�1B+D is notidentically zero. Under these conditions, the system (4.93) has for all initialconditions a unique piecewise di�erentiable right-Zeno solution if and only ifthe leading Markov parameter is positive.It is typical to �nd that well-posedness of complementarity systems is linkedto a positivity condition. If the number of pairs of complementary variables islarger than one, an appropriate matrix version of the positivity condition hasto be used. As might be expected, the type of positivity that we need is the\P-matrix" property from the theory of the LCP. Recall (see the end of theIntroduction of this chapter) that a square real matrix is said to be a P-matrixif all its principal minors are positive.To state a result on well-posedness for multivariable linear complementaritysystems, we again need some concepts form linear system theory. Recall (seefor instance [84, p. 384] or [89, p. 24]) that a square rational matrix G(s) issaid to be row proper if it can be written in the formG(s) = �(s)B(s) (4.109)where �(s) is a diagonal matrix whose diagonal entries are of the form sk forsome integer k that may be di�erent for di�erent entries, and B(s) is a properrational matrix that has a proper rational inverse (i. e. B(s) is bicausal). Aproper rational matrix B(s) = B0 +B1s�1+ � � � has a proper rational inverseif and only if the constant matrix B0 is invertible. This constant matrixis uniquely determined in a factorization of the above form; it is called theleading row coe�cient matrix of G(s). In a completely similar way one de�nes

4.4. Linear complementarity systems 103the notions of column properness and of the leading column coe�cient matrix.We can now state the following result [69, Thm. 6.3].Theorem 4.4.5. The linear complementarity system (4.93) is well-posed ifthe associated transfer matrix G(s) = C(sI�A)�1B+D is both row and columnproper, and if both the leading row coe�cient matrix and the leading columncoe�cient matrix are P-matrices. Moreover, in this case the multiplicity ofevents is at most one, i. e. at most one reinitialization takes place at timeswhen a mode change occurs.An alternative su�cient condition for well-posedness can be based on therational complementarity problem (RCP) that was already used above (section4.4). For a given set of linear system parameters (A;B;C;D), we denote byRCP(x0) the rational complementarity problem RCP(q(s),M(s)) with dataq(s) = C(sI � A)�1x0 and M(s) = C(sI � A)�1B +D. For the purposes ofsimplicity, the following result is stated under somewhat stronger hypothesesthan were used in the original paper [70, Thm. 5.10, 5.16].Theorem 4.4.6. Consider the linear complementarity system (4.93), and as-sume that the associated transfer matrix is totally invertible. The system (4.93)is well-posed if the problem RCP(x0) has a unique solution for all x0.A connection between the rational complementarity problem and the stan-dard linear complementarity problem can be established in the following way[70, Thm. 4.1, Cor. 4.10].Theorem 4.4.7. For given q(s) 2 Rk (s) and M(s) 2 Rk�k (s), the problemRCP(q(s);M(s)) is uniquely solvable if and only if there exists � 2 R suchthat for all � > � the problem LCP(q(�);M(�)) is uniquely solvable.The above theorem provides a convenient way of proving well-posedness forseveral classes of linear complementarity systems. The following example istaken from [70].Example 4.4.8. A linear mechanical system may be described by equationsof the formM �q +D _q +Kq = 0 (4.110)where q is the vector of generalized coordinates, M is the generalized massmatrix, D contains damping and gyroscopic terms, and K is the elasticitymatrix. The mass matrix M is positive de�nite. Suppose now that we subjectthe above system to unilateral constraints of the formFq � 0 (4.111)where F is a given matrix. Under the assumption of inelastic collisions, thedynamics of the resulting system may be described byM �q +D _q +Kq = F Tu; y = Fq (4.112)

104 Chapter 4. Complementarity systemstogether with complementarity conditions between y and u. The associatedRCP is the following:y(s) = F (s2M + sD +K)�1[(sM +D)q0 +M _q0]+ F (s2M + sD +K)�1F Tu(s): (4.113)If F has full row rank, then the matrix F (s2M + sD + K)�1F T is positivede�nite (although not necessarily symmetric) for all su�ciently large s becausethe term associated to s2 becomes dominant. By combining the standard resulton solvability of LCPs with Thm. 4.4.7, it follows that RCP is solvable and wecan use this to prove the well-posedness of the constrained mechanical system.This provides some con�rmation for the validity of the model that has beenused, since physical intuition certainly suggests that a unique solution shouldexist.In the above example, one can easily imagine cases in which the matrix Fdoes not have full row rank so that the ful�llment of some constraints alreadyimplies that some other constraints will also be satis�ed; think for instance of achair having four legs on the ground. In such cases the basic result on solvabil-ity of LCPs does not provide enough information, but there are alternativesavailable that make use of the special structure that is present in equations like(4.113). On the basis of this, one can still prove well-posedness; in particularthe trajectories of the coordinate vector q(t) are uniquely determined, eventhough the trajectories of the constraint force u(t) are not.4.5 Mechanical complementarity systemsMechanical systems with unilateral constraints can be represented as semi-explicit complementarity systems (cf. [138]):_q = @H@p (q; p) q 2 Rn ; p 2 Rn_p = �@H@q (q; p)� @R@ _q ( _q) + @CT@q (q)u u 2 Rky = C(q); y 2 Rk (4.114a)0 � y ? u � 0: (4.114b)The presentation here is the same as in (4.8) except that we have added aRayleigh dissipation function R. Assume that the system (4.114a) is real-analytic, and that the unilateral constraints are independent, that isrank @CT@q (q) = k; for all q with C(q) � 0: (4.115)Since the Hamiltonian is of the form (kinetic energy plus potential energy)H(q; p) = 12pTM�1(q)p+ V (q); M(q) =MT (q) > 0 (4.116)

4.5. Mechanical complementarity systems 105whereM(q) is the generalized mass matrix, it follows that the system (4.114a)has uniform relative degree 2 with decoupling matrixD(q) = �@CT@q (q)�T M�1(q)@CT@q (q): (4.117)Hence, from M(q) > 0 and (4.115) it follows that D(q) is positive de�nite forall q with C(q) � 0. Since the principal minors of a positive de�nite matrixare all positive, all conditions of Thm. 4.3.1 and Thm. 4.3.2 are satis�ed, andwe establish well-posedness for smooth continuations. We have essentiallyfollowed an argument in [96].A switch rule for mechanical complementarity systems can be formulatedas follows. Let us consider a mechanical system with n degrees of freedomq = (q1; � � � ; qn) having kinetic energy 12 _qTM(q) _q, where M(q) > 0 is the gen-eralized mass matrix. Suppose the system is subject to k geometric inequalityconstraintsyi = Ci(q) � 0; i 2 K = f1; � � � ; kg (4.118)If the i-th inequality constraint is active, that is Ci(q) = 0, then the systemwill experience a constraint force of the form @Ci@q (q)ui, where @Ci@q (q) is thecolumn vector of partial derivatives of Ci and ui a Lagrange multiplier.Let us now consider an arbitrary initial continuous state (q�; _q�). De�nethe vector of generalized velocitiesv� := @CI@q (q�) _q� (4.119)where I denotes the set of active indices at q�. In order to describe theinelastic collision we consider the system of equalities and inequalities (in theunknowns v+; �)v+ = v� + @CI@q (q�)M�1(q�)@CTI@q (q�)�0 � v+ ? � � 0: (4.120)Here � can be interpreted as a vector of Lagrange multipliers related to im-pulsive forces. The system (4.120) is in the form of a linear complementarityproblem (LCP). The general form of an LCP can be written asy = x+Mu; 0 � y ? u � 0 (4.121)where the vector x and the square matrix M are given, and the vectors y andu are the unknowns. As already noted in the Introduction of this chapter, itis a classical result that the LCP (4.121) has a unique solution (y; u) for eachx if and only if the matrix M is a P-matrix, that is to say, if and only if allprincipal minors of the matrix M are positive. This holds in particular if M

106 Chapter 4. Complementarity systemsis a positive de�nite matrix. Since @CI@q (q�)M�1(q�)@CTI@q (q�) > 0, the LCP(4.120) has a unique solution. The jump rule is now given by(q�; _q�) 7! (q+; _q+);with q+ = q�; _q+ = _q� +M�1(q�)@CTI@q (q�)�: (4.122)The new velocity vector _q+ may equivalently be characterized as the solutionof the quadratic programming problemminf _q+jC(q) _q+�0g 12 ( _q+ � _q�)TM(q)( _q+ � _q�) (4.123)where q := q� = q+. This formulation is sometimes taken as a basic principlefor describing multiple inelastic collisions; see [31, 112]. Note that there is asimple interpretation to the quadratic programming problem (4.123): in thetangent space at the con�guration q, the problems calls for the determination ofthe admissible velocity that is closest to the impact velocity, where \closest" isinterpreted in the sense of the metric given by the kinetic energy. An appealingfeature of the transition rule above is that the energy of the mechanical systemwill always decrease at the switching instant. One may take this as a startingpoint for stability analysis.Example 4.5.1 (Two carts with stop and hook). As an example of theswitching rule described above, let us again consider the two-carts system ofSubsection 2.2.9. To make things more interesting, we add a second constraintwhich might be realized as a hook (see Fig. 4.2). The equations of motion are�x1(t) = �2x1(t) + x2(t) + u1(t) + u2(t) (4.124a)�x2(t) = x1(t)� x2(t)� u2(t) (4.124b)y1(t) = x1(t) (4.124c)y2(t) = x1(t)� x2(t) (4.124d)0 � y1(t) ? u1(t) � 0 (4.124e)0 � y2(t) ? u2(t) � 0: (4.124f)Consider now the multiple-impact point (x1; x2) = (0; 0). The two-dimensional tangent space at this point contains several regions of interestwhich are indicated in Fig. 4.3. In the �gure, the horizontal axis is used forv1 := _x1 and the vertical axis for v2 := _x2. The cone of admissible post-impactvelocities is given by the conditions v1 � 0 and v1�v2 � 0; this region has beenlabeled A in the �gure. The opposite cone contains all possible pre-impact ve-locities, and consists of three regions which have been labeled B, C and D.According to the jump rule speci�ed above, a given pre-impact velocity willbe mapped to the post-impact velocity that is closest to it in the sense of the

4.5. Mechanical complementarity systems 107������

������ Figure 4.2: Two carts with stop and hook

A

Dv

v1

2

C

BFigure 4.3: Tangent planekinetic metric, which in this case is just the standard Euclidean metric sincethe masses of both carts are assumed to be 1. (We leave it to the reader towork out the jump condition in other cases, for instance when the mass of theright cart is twice the mass of the left cart.) Therefore, a pre-impact velocityin region B will be mapped to a post-impact velocity on the hal ine v1 = 0,v2 � 0, so that the left cart will remain in contact with the stop whereas thehook contact is not active. For pre-impact velocities in region C, the origin isthe closest point in the cone of admissible velocities. This means that if thepre-impact velocities are such that v1 � 0 and 0 � v2 � �v1, the impact willbring the system to an instantaneous standstill. Finally if v1 � 0 and v2 � �v1just before impact (region D), then the hook contact will be maintained, sothe two carts remain at a constant distance, whereas the contact at the stop isreleased (except in the trivial case in which the pre-impact velocities are zero,so that in fact there is no impact at all).Since the system in our example is linear, we may also treat it by themethods of Section 4.4; in particular we may apply the RCP-based switchingrule (4.103). In the present case, the rational complementarity problem (4.102)takes the following form, for general initial conditions xi(0) = xi0, _xi(0) = vi0

108 Chapter 4. Complementarity systems(i = 1; 2):(s4 + 3s2 + 1)y1(s) == (s2 + 1)sx10 + sx20 + (s2 + 1)v10 + v20 ++ (s2 + 1)u1(s) + s2u2(s) (4.125a)(s4 + 3s2 + 1)y2(s) == s3x10 � (s2 + 1)sx20 + s2v10 � (s2 + 1)v20 ++ s2u1(s) + (2s2 + 1)u2(s): (4.125b)We are interested in particular in the situation in which x10 = 0 and x20 = 0.To �nd out under what conditions the rule (4.103) allows for instance a jumpto the stop-constrained mode, we have to solve the above rational equations fory2(s) and u1(s) under the conditions y1(s) = 0 and u2(s) = 0. The resultingequations are 0 = (s2 + 1)v10 + v20 + (s2 + 1)u1(s) (4.126a)(s4 + 3s2 + 1)y2(s) = s2v10 � (s2 + 1)v20 + s2u1(s): (4.126b)These equations can be readily solved, and one obtainsy2(s) = � 1s2 + 1v20 (4.127a)u1(s) = �v10 � 1s2 + 1v20: (4.127b)These functions are nonnegative in the ordering we have put on rational func-tions if and only if v10 � 0 and v20 � 0; so it follows that the jump to thestop-constrained mode is possible according to the RCP rule if and only ifthese conditions are satis�ed. Moreover, the re-initialization that takes placeis determined by (4.127b) and consists of the mapping (v10; v20) 7! (0; v20).Note that these results are in full agreement with the projection rule basedon the kinetic metric. This concludes the computations for the jump to thestop-constrained mode. In the same way one can verify that actually in allcases the RCP rule leads to the same results as the projection rule. It canbe proved for linear Hamiltonian complementarity systems in general that theRCP rule and the projection rule lead to the same jumps; see [69].It should be noted that in the above example we consider only some ofthe �rst elements of impact theory. In applied mechanics one needs to dealwith much more complicated impacts in which also frictional e�ects and elasticdeformations may play a role. In any given particular situation, one needs tolook for impact models that contain enough freedom to allow a satisfactorydescription of a range of observed phenomena, and that at the same time arereasonably identi�able in the sense that parameter values can be obtained to

4.6. Relay systems 109good accuracy from experiments. In addition to the modeling problems, onealso faces very substantial computational problems in situations where thereare many contact points, such as for instance in the study of granular material.Remark 4.5.2. If in the example above one replaces the initial data(x10; x20) = (0; 0) by (x10; x20) = ("; 0) or (x10; x20) = (0;�"), then oneobtains quite di�erent solutions. The system in the example therefore dis-plays discontinuous dependence on inital conditions . We see that such a phe-nomenon, which is quite rare for ordinary di�erential equations, occurs alreadyin quite simple examples of hybrid dynamical systems. This again indicatesthat there are fundamental di�ences between smooth dynamical systems andhybrid dynamical systems; we already noted before that in simple examples ofhybrid systems one may have right uniqueness of solutions but no left unique-ness, which is also a phenomenon that normally doesn't occur in systemsdescribed by ordinary di�erential equations.The discontinuous dependence on initial conditions may be viewed as aresult of an idealization, re ecting a very sensitive dependence on initial con-ditions in a corresponding smooth model. Certainly the fact that such discon-tinuities appear is a problem in numerical simulation, but numerical problemswould also occur when for instance the strict unilateral constraints in the ex-ample above would be replaced by very sti� springs. So the hybrid modelin itself cannot be held responsible for the (near-)discontinuity problem; oneshould rather say that it clearly exposes this problem.4.6 Relay systemsFor piecewise linear relay systems of the form_x = Ax+Bu; y = Cx+Du; ui = � sgnyi (i = 1; : : : ; k) (4.128)one may apply Thm. 4.4.7, but the application is not straightforward for thefollowing reason. As noted above, it is possible to rewrite a relay systemas a complementarity system (in several ways actually). Using the method(4.29), one arrives at a relation between the new inputs col(u1; u2) and thenew outputs col(y1; y2) that may be written in the frequency domain as follows(1 denotes the vector all of whose entries are 1, and G(s) denotes the transfermatrix C(sI �A)�1B +D):24 u1(s)u2(s) 35 = 24 �G�1(s)C(sI �A)�1x0 + s�11G�1(s)C(sI �A)�1x0 + s�11 35++ 24 G�1(s) �G�1(s)�G�1(s) G�1(s) 3524 y1(s)y2(s) 35 : (4.129)The matrix that appears on the right hand side is singular for all s and sothe corresponding LCP does not always have a unique solution. However the

110 Chapter 4. Complementarity systemsexpression at the right hand side is of a special form and we only need toensure existence of a unique solution for LCPs of this particular form. On thebasis of this observation, the following result is obtained.Theorem 4.6.1. [95, 70] The piecewise linear relay system (4.128) is well-posed if the transfer matrix G(s) is a P-matrix for all su�ciently large s.In particular the theorem may be used to verify that the system in the exampleat the beginning of Section 4.3, with the \right" sign of the relay feedback, iswell-posed.The above result gives a criterion that is straightforward to verify (computethe determinants of all principal minors of G(s), and check the signs of theleading Markov parameters), but that is restricted to piecewise linear systems.Filippov [54, x2.10] gives a criterion for well-posedness which works for generalnonlinear systems, but needs to be veri�ed on a point-by-point basis.4.7 Notes and references for Chapter 4Complementarity problems have been studied intensively since the mid-sixties,and much attention has in particular been given to the linear complementar-ity problem. The book [39] by Cottle, Pang, and Stone provides a rich sourceof information on the LCP. The general formulation of input-output dynami-cal systems as in (4.4a{4.4b) has been popular in particular in control theorysince the early sixties. See for instance [121] for a discussion of nonlinearsystems, and [84] for linear systems. The combination of complementarityconditions and di�erential equations has been used for mechanical problemsby L�otstedt [96]; related work has been done by Moreau who used a some-what di�erent formulation (the \sweeping process") [113]. In the mechanicalcontext, complementarity conditions have not only been used for the descrip-tion of unilateral constraints but also for the modeling of dry friction; see forinstance [129, 146, 145]. Another area where the combination of complemen-tarity conditions and di�erential equations arises naturally is electrical circuitsimulation; early work in the simulation of piecewise linear electrical networkshas been done by Van Bokhoven [23]. The idea of combining complementarityconditions with general input-output dynamical systems seems to have beenproposed �rst in [138].2 Further information about complementarity systemsis available in [33, 34, 68, 69, 70, 71, 72, 73, 95, 139, 140, 141].2In the cited paper the term \complementary-slackness system" was used. In later workthis has been changed to \complementarity system" because this term is shorter and connectsmore closely to the complementarity problem, and also because the English word slacknessseems to be hard to translate into other languages such as Dutch.

Chapter 5Analysis of hybrid systemsTypical properties studied for smooth dynamical systems include the nature ofequilibria, stability, reachability, the presence of limit cycles, and the e�ects ofparameter changes. In the domain of �nite automata, one may be interestedfor instance in the occurrence of deadlock, the correctness of programs, inclu-sion relations between languages recognized by di�erent automata, and issuesof complexity and decidability. In the context of hybrid systems, one may ex-pect to encounter properties both from the continuous and from the discretedomain, with some interaction between them. In this chapter we discuss anumber of examples.5.1 Correctness and reachabilityOver the past three centuries, the investigation of properties of smooth dynam-ical systems has been a subject of intense research. Ingenious methods havebeen used to obtain many results of interest and of practical relevance, but ofcourse the subject is too wide to be ever completely understood. The typicalapproach is to look for properties that can be proven to hold for suitably de-�ned subclasses; there is no general proof method, and usually considerableingenuity is needed to arrive at interesting conclusions. In the theory of �niteautomata however we encounter a situation that is radically di�erent. Sincethe number of states is �nite, certain properties can be veri�ed in principleby checking all possible states. The catch is in the words \in principle"; thenumber of states to visit may indeed be huge in situations of practical interest,and so one should be concerned with the complexity of algorithms and variousheuristics.5.1.1 Formal veri�cationThe term formal veri�cation (or computer-aided veri�cation) is used for themethods that have been developed in computer science to analyze the prop-erties of discrete systems. Of particular interest is to prove the correctnessof programs (understood in a wide sense to include software but also com-munication protocols and hardware descriptions). To �nd bugs in a complexsoftware system is a highly nontrivial matter, as is well-known, and given the111

112 Chapter 5. Analysis of hybrid systemssize of the software industry there is a major interest in automated tools thatcan help in the veri�cation process.There are two types of formal veri�cation methods, known as theorem prov-ing and model checking . In theorem proving, one starts with a representationof the program to be analyzed and attempts to arrive at a proof of correct-ness by applying a set of inference rules. Theorem provers often need somehuman guidance to complete their task in a reasonable amount of time. More-over, when the proof of correctness fails, a theorem prover often provides littleindication of what is wrong in the design. Model checking is in principle abrute-force approach, building on the �niteness of the state space. By search-ing all possibilities, the method either proves a design to be correct or �nds acounterexample. The ability of model checkers to �nd bugs is of course veryimportant to system designers. The time needed to do an exhaustive searchis however exponential in terms of the number of variables in the system, andso considerable attention has gone into the development of methods that cancombat complexity.Program properties to be speci�ed are often expressed in terms of temporallogic formulas; these are able to express properties like \if the variable P isreset to zero, then eventually the variable Q is reset to zero as well". More gen-erally, correctness can be viewed as an implementation relation in which thespeci�cations are expressed in some language, the design is expressed in pos-sibly a di�erent language, and it is to be proven that the design \implements"the speci�cation. There can actually be several stages of such relations; thenone has a hierarchy of models, each of which is proved to be an implementationof the one on the next higher level. A structure of this type is obtained by thedesign method of stepwise re�nement , in which one starts on an abstract levelthat allows relatively easy veri�cation, and then proceeds in steps to more andmore re�ned designs which are veri�ed to be speci�c ways of carrying out theoperations on the previous level. Conversely, given a fully speci�ed model, onecan try to simplify the model for the purpose of checking a particular propertyby leaving out detail that is believed to be irrelevant in connection with thatproperty. Typically one then gets a nondeterministic model, in which a pre-cise but complicated description of a part of the system is replaced by somecoarse information which nevertheless may still be enough to establish the de-sired property. This process is sometimes called abstraction. Some reductionof complexity may also be achieved by choosing suitable representations ofsets of states; in particular it is often useful to work with Boolean expressionswhich can describe certain sets of states in a compact way. Methods that usesuch expressions come under the heading of symbolic model checking . In ad-dition to model checkers there are also equivalence checkers which verify thecorrectness of a design after modi�cation on the assumption that the originaldesign was correct. Of course the feasibility of various methods to overcomethe complexity barrier also depends on the way that programs are formed, andin this context the idea of modular programming is important.A general caveat that should be kept in mind is that formal veri�cation

5.1. Correctness and reachability 113can never give absolute certainty about the functioning of a piece of computerequipment. The reason for this is that veri�cation methods operate on a formalmodel, which relates to the real world through a formalization step that maycontain mistakes. For instance, if a design is formally proved correct but theimplementation routine which takes the design to chip circuitry contains anerror, then the result may still be faulty. Reportedly this is what happened inthe case of the infamous Pentium division bug.In spite of such incidents, formal veri�cation methods and in particularmodel checking constitute one of the success stories of computer science, es-pecially in the area of veri�cation of hardware descriptions. All of the majorcomputer �rms have developed their own automated veri�cation tools, andcommercial tools are �nding their way to the marketplace. It is no surprise,therefore, that a large part of the interest of computer scientists who workwith hybrid systems goes to veri�cation. Although a general formulation ofcorrectness for hybrid systems should be in terms of language inclusion (andone might discuss which languages are suitable for this purpose), certain prop-erties of interest can be expressed more simply as reachability properties, sothat the legal trajectories are those in which certain discrete states are notreached. This is illustrated in the next subsection.5.1.2 An audio protocolExample 5.1.1. A certain protocol for communication between the subsys-tems of a consumer audio systems was suggested as a benchmark in [24]. Theproblem is to verify that the protocol is correct in the sense that is ensuresthat no communication errors occur between the subsystems. Without goinginto the details of the actual implementation (see [24] for a more elaboratedescription), the protocol can be described as follows. The purpose of theprotocol is to allow the components of a consumer audio system to exchangemessages. A certain coding scheme is used which depends on time intervalsbetween events, an \event" being the voltage on a bus interface going fromlow to high. A basic time interval is chosen and the coding can be describedas follows:- the �rst event always signi�es a 1 (messages always start with a 1);- if a 1 has last been read, the next event signi�es 1 if 4 basic time intervalshave passed, 0 if 6 intervals have passed, and 01 if 8 intervals have passed;- if a 0 has last been read, the next event signi�es 0 if 4 intervals havepassed, and 01 if 6 intervals have passed;- if more than 9 intervals pass after a 1 has been read, or more than 7after a 0 has been read, the message is assumed to have ended.Due to clock drift and priority scheduling, the timing of events is uncertainand the design speci�cations call for a 5% tolerance in timing. Clocks are resetwith each event.

114 Chapter 5. Analysis of hybrid systemsTo describe the protocol using event- ow formulas, it is convenient to modelthe sender and the receiver separately. Let us begin with the description of thesender. We use two continuous variables, namely clock time denoted by xs andan o�set signal denoted by us. There are two discrete state variables As andLs. The variable As takes the value 1 if currently a message is being sent and 0otherwise. The variable Ls denotes the last symbol that has been sent, whichis either 0 or 1. There is also a discrete communication variable denoted by Ssrepresenting the signal to be sent to the receiver; this signal is prescribed fromoutside in a way that is not modeled. Because in the de�nition of event- owformulas we have taken as a default that events are not synchronized, we mustalso introduce a communication variable between sender and receiver whichwe denote by C. Technically this corresponds to the fact that both sender andreceiver are aware of voltage surges on the connecting bus. The operation ofthe sender can now be described as follows:sender : clock�sender jj events�sender (5.1a)clock�sender : _xs = 1 + us; �0:05 � us � 0:05 (5.1b)events�sender : C = 1; x+s = 0; ��������� A�s = 0; wakeup�sA+s = 1; ������ L�s = 0; event0�sL�s = 1; event1�s(5.1c)wakeup�s : Ss = 1; A+s = 1; L+s = 1 (5.1d)event0�s ������������ xs = 4; Ss = 1; L+s = 1xs = 6; Ss = 0; L+s = 0xs = 8; Ss = 01; L+s = 1xs � 9; A+s = 0 (5.1e)event1�s ��������� xs = 4; Ss = 0; L+s = 0xs = 6; Ss = 01; L+s = 1xs � 7; A+s = 0: (5.1f)Now we model the receiver. While the sender acts at integer time points asmeasured on its own clock, the events do not always take place after an integernumber of time units on the receiver's clock, as a result of clock inaccuracies.Let us assume that the receiver will round times to the nearest integer in theset f4; 6; 8; 9g. The operation of the receiver may then be described as follows,notation being similar to the one used for the sender:receiver : clock�receiver jj events�receiver jj timelimit (5.2a)clock�receiver : _xr = 1 + ur; �0:05 � ur � 0:05 (5.2b)

5.1. Correctness and reachability 115events�receiver : C = 1; x+r = 0; ��������� A�r = 0; wakeup�rA+r = 1; ������ L�r = 0; event0�rL�r = 1; event1�r(5.2c)wakeup�r : Sr = 1; A+r = 1; L+r = 1 (5.2d)event0�r ������������ xr < 5; Sr = 1; L+r = 15 � xr < 7; Sr = 0; L+r = 07 � xr < 9; Sr = 01; L+r = 1xr = 9; A+s = 0 (5.2e)event1�r ��������� xr < 5; Sr = 0; L+r = 05 � xr < 7; Sr = 01; L+r = 1xr = 7; A+r = 0 (5.2f)timelimit : Lr = 1; xr � 9 j Lr = 0; xr � 7: (5.2g)We consider solutions in NZ/1/C1/C0 of the system as a whole, which isdescribed bysystem : sender jj receiver: (5.3)In particular we are interested in the question whether the discrete externalcommunication traces of sender and receiver are identical for all solutions.The protocol is said to be correct if the string of output symbols is alwaysequal to the string of input symbols. The correctness condition can be de-scribed as a (discrete) reachability condition: no discrete states (s1; s2) withs1 6= s2 should be reachable. This means that transitions to such states shouldnever be possible. The veri�cation of this condition requires inspection of thejump conditions and the transition rules, and since these involve continuousdynamics, we have at each discrete state and for each discrete input value acontinuous reachability problem. Actually in the present case the continuousdynamics is the same at each discrete state, so it su�ces to draw a singlepicture. In Fig. 5.1 the reachable set of continuous states (x1; x2) is indicated(shaded) together with the set of points that should be avoided in order toprevent illicit transitions. The reachable set is a cone whose width is deter-mined by the tolerance of the clocks; it is seen that the 5% tolerance is enough(although only barely so) to ensure the correctness of the protocol. An in-teresting feature of the example is that although as far as the description ofthe dynamics is concerned there would be no need to distinguish for instancebetween the discrete states (1; 1) and (1; 01), this distinction does howeverbecome important if one wants to formulate correctness requirements. Theexample also shows that such requirements may take the form of reachabilityconditions that are ultimately formulated in the continuous state space.

116 Chapter 5. Analysis of hybrid systems

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Figure 5.1: Reachable set and avoidance set for audio protocol

5.1. Correctness and reachability 1175.1.3 Algorithms for veri�cationGiven that veri�cation problems can often be written as reachability problems,the question arises whether it is possible in principle to write an algorithmthat will decide, given a hybrid system and an initial condition, whether acertain state is reachable or not. For �nite automata, it is easy to constructsuch an algorithm, so that the discussion in this case would concentrate onthe e�ciency of algorithms. In the case of continuous systems on the otherhand, a simple test is available for the case of linear �nite-dimensional time-invariant systems, but there is no general test for nonlinear systems and itwould be too much to expect that such a test can be found. So along a scalethat begins with �nite automata and ends with general hybrid systems, thereshould be somewhere a point that divides the system types that can be testedfor reachability from those for which this is not the case.It has been shown that for timed automata the reachability problem isdecidable [1]. Timed automata can be looked at as hybrid systems in whichthere is only one continuous variable called \time", which satis�es the dif-ferential equation _t = 1. The inclusion of this variable makes it possible toexpress quantitative statements (\after the command login, connection tothe host computer takes place within three seconds") rather than only qual-itative statements such as in standard temporal logic (\after the commandlogin, connection to the host computer takes place eventually"). From thestandpoint of �nite automata this is a major extension, from the standpointof hybrid systems however timed automata form a rather small class.It was shown in [1] that the reachability problem is still decidable for hy-brid systems that have several clocks running at di�erent rates (so continuousvariables xi satisfying di�erential equations of the form _xi = ci where the ci'sare constants), as long as all inequality constraints involving these clocks areof the form xi � k or xi � k (so not of the form xi � xj). As soon as oneallows two clocks that run at di�erent rates and that may be compared to eachother, however, the reachability problem becomes undecidable [1, Thm. 3.2].So it must be concluded that, for hybrid systems in which there is any seri-ous involvement of continuous variables, one cannot expect to �nd a generalalgorithm that will decide reachability.This being the case, one can still attempt to design algorithms that willsometimes terminate, in particular for limited classes of hybrid systems. Thetool HyTech [76] has been built at Cornell for the reachability analysis ofsystems with several clocks under constraints that guarantee that the reachablesets in the continuous state space can always be described by sets of linearinequalities. A symbolic model checker for timed automata is Kronos [41],developed at IMAG in Grenoble.So what can be said about the veri�cation of hybrid systems that involvea signi�cant amount of (nonlinear) continuous dynamics? The problem itselfis certainly of importance; there are many cases in which computer programsinteract with the continuous \real" world, and it can literally be a matter of lifeand death to make sure that the interaction takes place as expected. It should

118 Chapter 5. Analysis of hybrid systemsbe noted in the �rst place that tools like HyTech can still be used for theanalysis of hybrid systems that do not strictly satisfy their assumptions, sinceone can approximate the given system by one that does fall within the scopeof the tool. If the approximation is done conservatively (so that for instancethe reachable set of the approximating system always contains the reachableset of the given system), one can even obtain strict guarantees. In the case oftools that are based on di�erential equations with constant right hand sides(so that the reachable sets are indeed polyhedral) one will need to approximatea given function by a piecewise constant function. Depending on the natureof the given function, a reasonable approximation may require a large numberof pieces, and both the memory requirements and the computation time of aformal veri�cation tool may soon become prohibitive.An alternative is to resort to simulation: simply generate a large num-ber of scenarios to see if any faults occur. This methodology can provide atleast some test of validity in cases where other methods do not apply. Purelyrandom simulation may take a long time to �nd potentially hazardous situa-tions; therefore one should use rare-event simulation techniques which shouldbe guided as much as possible by information about situations where problemsmay be expected. Such information might come for instance from model check-ing on a coarse approximation of the system. Alternatively one may attemptto look systematically for worst-case situations, by introducing an \adversary"who is manipulating disturbance inputs (or more generally any available non-determinism) with malicious intent. The adversary's behavior may be formallyobtained as the solution of an optimization problem which may however bedi�cult to solve; in general, optimization problems in a hybrid system contexthave not been studied much yet. If at the same time one wants to design opti-mal strategies for the system to respond to adverse circumstances one obtainsa minimax game problem, which may be even harder to solve. Experiencein case studies such as the one in [88] will have to determine the ranges ofapplicability of each of the possible methodologies or combinations of them.5.2 Stability5.2.1 Lyapunov functions and Poincar�e mappingsIn the study of the stability of nonlinear dynamical systems, Lyapunov func-tions and Poincar�e mappings play a central role. Lyapunov functions canbe used to prove the stability of equilibria or the invariance of certain sets;such sets are the level sets of a scalar function on the state space which hasmonotonous behavior along system trajectories. The idea behind Poincar�emappings is that of the stroboscope. By looking at the continuous states onlyat certain points in time, one obtains a discrete-time system. By analyzingthis discrete-time system, one may obtain information about the stability ofcertain motions of the continuous-time system (for instance periodic orbits).For nonlinear systems there are in fact many di�erent notions of stability,

5.2. Stability 119ranging from local stability to uniform global asymptotic stability. The notionof stability may be applied to equilibrium points but also more generally toinvariant sets. Given a dynamical system, a subset M of the state space issaid to be (forward) invariant if all trajectories that have their initial point inM are in fact full contained in M . With this terminology, an equilibrium maysimply be de�ned as an invariant set consisting of one point.In the hybrid context, we have a state space with a continuous and adiscrete part. There is no problem in extending the notion of invariance to thissetting. Convergence to a discrete equilibrium point means that the systemeventually settles in one particular mode; such a property may be of interestin speci�c applications. Certainly also the notion of invariance of a certainsubset of the discrete state space can be of importance, for instance froma veri�cation point of view. Methods that allow proofs of convergence andinvariance are therefore at least as important in hybrid systems as they are insmooth dynamical systems.First let us review a few de�nitions that are used in stability theory. Con-sider a dynamical system de�ned on some state space X which is equippedwith some metric d. The state space X may be a product of a continuous spacelike Rk (or more generally a di�erentiable manifold) and a discrete space S.The set S of possible values of the discrete variables will always be assumed tocarry the discrete metric, which means that convergence is the same as even-tual equality. If one is interested only in stability of the continuous variables,one may consider the dynamical system de�ned on the continuous part whichis obtained by simply omitting the symbolic parts of trajectories. Now let x0be an equilibrium point. The equilibrium is said to be stable (or sometimesalso stable in the sense of Lyapunov , or marginally stable) if for every " > 0there exists a � > 0 such that every trajectory that starts at a distance lessthan � from x0 will stay within a distance " from x0. In other words, ac-cording to this de�nition x0 is stable if we can guarantee that trajectories willremain arbitrarily close to x0 by giving them an initial condition su�cientlyclose to x0. There is no implication here that trajectories will converge tox0. In the following notion of stability we do have such an implication. Theequilibrium x0 is said to be asymptotically stable if there exists " > 0 such thatall trajectories with initial conditions at distance less than " to x0 convergeto x0. Clearly this is a local notion of stability. The equilibrium x0 is saidto be globally asymptotically stable if all trajectories of the system convergeto x0. There are various related notions such as ultimate boundedness andexponential stability that we shall not discuss here; also, all de�nitions maybe generalized in a straightforward way to the case in which the equilibriumx0 is replaced by an invariant set.In the standard application of Lyapunov functions to smooth dynamicalsystems, the stability of an equilibrium can be concluded if it possible to�nd a continuous scalar function V (x) that has a minimum at x0 and thatis nonincreasing along trajectories. By imposing additional conditions on theLyapunov function V (x), one can obtain stronger properties such as asymp-

120 Chapter 5. Analysis of hybrid systemstotic stability and global asymptotic stability. At �rst sight it would seem thatthe condition that V (x) should be nonincreasing along trajectories depends onknowing the solutions. Of course, any method that requires the computation ofsolutions is uninteresting, �rstly because it is usually not possible to computethe solutions explicitly, and secondly because the stability problem becomestrivial once the solutions are known (because one can then see immediatelyfrom the computed trajectories whether stability holds). Fortunately, if V (x) isdi�erentiable and the system is given by a di�erential equation _x(t) = f(x(t)),one can verify that V (x) is nonincreasing along trajectories, without knowingthe solutions, by checking that (@V=@x)(x)f(x) � 0. In the context of hybridsystems however this simple check is not available, since the system is notgiven by a single di�erential equation and there may be jumps in the statetrajectories at event times. For this reason the transplantation of Lyapunovtheory from the continuous to the hybrid domain is not straightforward.The most direct generalization of the method of Lyapunov functions tohybrid systems would be the following (see for instance [161]). Suppose thereis a continuous function V (x) de�ned on the continuous state space which hasa minimum at an equilibrium point x0, is nondecreasing along trajectories oninterevent intervals, and satis�es V (x+) � V (x�) whenever x� and x+ areconnected through a jump rule. Under these conditions, the equilibrium x0is stable. With appropriate additional conditions on V (x), stronger conclu-sions can be drawn; also, analogous statements can be made for invariant setsinstead of equilibrium points. As is the case for smooth nonlinear systems,there are no general rules for constructing functions that satisfy the aboveproperties. For hybrid systems in which jumps occur, the condition that V (x)should also be nonincreasing across jumps may introduce an additional com-plication. An example in which a Lyapunov function is easy to �nd is the classof unilaterally constrained mechanical systems with Moreau's switching rulefor inelastic collisions, where the energy can be taken as such (see Section 4.5).In some cases, information from the event conditions may make it easier toconstruct a Lyapunov function. For instance, in the case of the bouncing ball(Subsection 2.2.3) it is natural to take the energy as a Lyapunov function, andit follows immediately from the event conditions that V (x+) = e2V (x�) atevent times, so that stability (even global asymptotic stability) of the equilib-rium follows if the restitution coe�cient e is less than 1. In between events, theenergy is constant and so without the information from the event conditionsone could only conclude stability in Lyapunov's sense. In general, stability ofan equilibrium point will follow if one can �nd a scalar function V satisfyingsuitable growth conditions and having a minimum 0 at x0, which is nonin-creasing on the sequence �1; �2; � � � of event times and for which an inequalityof the form V (x(t)) � h(V (x(�k))) (t 2 (�k; �k+1)) can be proven, where h isa nonnegative function satisfying h(0) = 0 [161, 127].Given the extremely wide scope of the class of hybrid dynamical systems,perhaps not too much should be expected from studies of stability that areconducted on this level of generality; for speci�c classes, it might be easier

5.2. Stability 121to come up with veri�able stability criteria. For systems that move betweenseveral operating conditions each related to a particular region of the statespace it may be a good idea to look for multiple Lyapunov functions de�nedon each of the operating regions. For systems that can be written as feedbacksystems consisting of a \plant" satisfying passivity conditions and a staticfeedback the hyperstability theory due to Popov [130] may be useful; here theproofs also ultimately depend on Lyapunov functions, but as a result of thespecial assumptions it is possible to give speci�c recipes for the construction ofthese functions. In situations in which one has design freedom for instance inthe choice of a control scheme, a possible approach is to try to ensure stabilityby choosing some function and making it a Lyapunov function by a suitablechoice of the design parameters.Stability of equilibria is of interest for systems that are supposed to stayclose to some operating point. Many systems however operate in a periodicmanner, and in that case one is interested in the stability of periodic orbits.An often used instrument in this context is the Poincar�e map (sometimes alsocalled return map). In the case of smooth dynamical systems, the idea of thereturn map works as follows. Given some periodic orbit that is to be checkedfor stability, take a point on the orbit and a surface through that point thatis transversal to the orbit. Trajectories that start on the surface su�cientlyclose to the chosen point will intersect the surface again after a time that isapproximately equal to the period of the orbit under investigation. The mapwhich takes the initial point of the trajectory to the point where it intersectsthe chosen surface again is called the Poincar�e map or return map. It is map-ping de�ned on a neighborhood of the point that we started with; one couldlook at the map as de�ning a discrete-time dynamical system on a space whosedimension is one lower than that of the original state space. It is in general anonlinear system, but the point on the orbit that we selected is an equilibrium(because of the periodicity of the orbit) and so we can linearize the systemaround this equilibrium by taking the Jacobian of the return mapping. If thelinearized system is stable (i. e. all eigenvalues are inside the unit circle), thenwe know that the periodic orbit is attracting so that the corresponding peri-odic regime is stable. Note that the method requires the computation of theJacobian of the return map, which may be done on the basis of a linearizationof the system in a neighborhood of the periodic orbit. The computation maynot be very easy, but at least it does not require �nding the solutions of thefull original nonlinear system.In the context of hybrid systems, the idea of using a return map mayeven be more natural than in the smooth context, since any switching surfacesthat occur provide natural candidates for the transversal surfaces on which thereturn map is de�ned. One has to be careful however since a small perturbationof an initial condition will in general have an e�ect on event times or might evenchange the order of events. Especially in cases where the dynamics betweenevents is fairly simple (for instance in piecewise linear systems), the Poincar�emap can nevertheless be a very e�cient tool.

122 Chapter 5. Analysis of hybrid systems5.2.2 Time-controlled switchingIf a dynamical system is switched between several subsystems, the stabilityproperties of the system as a whole may be quite di�erent from those of thesubsystems. This can already be illustrated in switching between two linearsystems, as is shown by the following calculation.Consider a system that follows the dynamics _x(t) = A1x(t) for a period 12",then switches to _x(t) = A2x(t) for again a period 12", then switches back, andso on. An event- ow formula for the system can be written down as follows:�������������������������� _� = 1; � � 12"; ������ P = 1; _x = A1xP = 2; _x = A1x�� = 12"; �+ = 0; ������ P� = 1; P+ = 2P� = 2; P+ = 1 (5.4)Let us consider a time point t0 at which the system begins a period in mode 1,with continuous initial state x0. At time t0+ 12", the state variable has evolvedto x(t0 + 12") = exp( 12"A1)x0 = x0 + "2A1x0 + "28 A21x0 + � � �(this is a consequence of the general rule exp(A) =P1k=0(1=k!)Ak). At timet0 + ", we getx(t0 + ") = (I + "2A2 + "28 A22 + � � � )(I + "2A1 + "28 A21 + � � � )x0= (I + "[ 12A1 + 12A2] + "28 [A21 +A22 + 2A2A1] + � � � )x0:If we compare the above expression to the power series development forexp["( 12A1 + 12A2)] which is given byexp["( 12A1 + 12A2)] = I + "[ 12A1 + 12A2] + "28 [A21 +A22 +A1A2 +A2A1] + � � �we see that the constant and the linear term are the same, whereas thequadratic term is o� by an amount of A1A2 � A2A1 (the \commutator" ofA1 and A2). So the di�erence between the solution of the switched systemand that of the smooth system _x = ( 12A1+ 12A2)x on a time interval of length" is of the order "2. If we let " tend to zero, then on any �xed time intervalthe solution of (5.4) will tend to the solution (with the same initial condition)of the \averaged" system_x(t) = ( 12A1 + 12A2)x(t): (5.5)In particular, the stability properties of the switched system (5.4) will for small" be determined by the stability properties of the averaged system (5.5), that

5.2. Stability 123is, by the location of the eigenvalues of 12A1+ 12A2. Now it is well-known thatthe eigenvalues are nonlinear functions of the matrix entries and so it may wellhappen that the matrices A1 and A2 are both Hurwitz (all eigenvalues havenegative real parts) whereas the matrix 12A1 + 12A2 is unstable, or vice versa.This is illustrated in the following example.Example 5.2.1. Consider the system (5.4) withA1 = 24 �0:5 1100 �1 35 ; A2 = 24 �1 �100�0:5 �1 35 : (5.6)Although A1 and A2 are both unstable, the matrix 12 (A1 + A2) is Hurwitz.Therefore the switched system should be stable if the frequency of switchingis su�ciently high. The switching frequency that is minimally needed can befound by computing the eigenvalues of the mapping exp( 12"A1) exp( 12"A2) asa function of "; stability is achieved when both eigenvalues are inside the unitcircle. For the present case, it turns out that a switching frequency of at least50 Hz is needed. The plot in Fig. 5.2 shows a trajectory of the system when itis switched at 100 Hz.

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Figure 5.2: Trajectory of the switched system de�ned by (5.6). Initial pointmarked by a star. Switching frequency 100 Hz; simulation period 2 units oftime

124 Chapter 5. Analysis of hybrid systemsInstead of staying both in mode 1 and in mode 2 for a time interval oflength 12", we can also consider the situation in which mode 1 is followedduring a time h" and mode 2 during a time interval (1 � h)", where h is anumber between 0 and 1. By the same reasoning as above, the behavior ofsuch systems on �xed time intervals is well approximated by that of the system_x(t) = Ahx(t), where Ah = hA1 + (1 � h)A2. The choice of the parameterh in uences the system dynamics and so h might be considered as a controlinput; the averaging analysis requires, however, that if h varies it should beon a time scale that is much slower than the time scale at which the switchingtakes place. When mode 1 corresponds to \power on" and mode 2 is \powero�", the parameter h is known as the duty ratio. In power electronics theuse of switches is popular because theoretically it provides a possibility toregulate power without loss of energy. Fast switching is a particular form ofthis method of regulation; for further discussion see 6.2.2.Remark 5.2.2. Using a standard trick in di�erential equations, one may con-sider the time parameter t as a state variable satisfying the simple di�erentialequation _t = 1. The systems considered in this subsection may then be seen asparallel compositions of two hybrid systems which are coupled via a discretecommunication channel. For instance, for the case of equidistant switchinginstants one may write an event- ow formula as follows:system = top-level jj low-level (5.7a)with top-level ������ _t = 1; t � 12"t� = 12"; t+ = 0; S = toggle (5.7b)low-level ��������� _x = APxS = toggle; ������ P� = 1; P+ = 2P� = 2; P+ = 1: (5.7c)Clearly the top-level mechanism might be replaced by something considerablymore complicated, which would lead us into largely unexplored research terri-tory. Stability results are available, however, for systems of the above type inwhich the top-level mechanism is a �nite-state Markov chain and the low-levelsystems are linear (jump linear systems); see for instance [107, 52].5.2.3 State-controlled switchingIn the previous subsection we have considered systems in which the switchingis not at all in uenced by the continuous state whose stability we are inter-ested in. Let us now take an opposite view and consider systems in which theswitching is completely determined by the continuous state. Below we shall

5.2. Stability 125consider in particular cases in which the discrete state of the system is deter-mined by the current continuous state, so that there is no \discrete memory"or \hysteresis". For such systems the continuous state space can be thought ofas being divided into cells which each correspond to a particular discrete state,so that each cell has its own continuous dynamics associated to it. Switchingof this type is often referred to as state-controlled , with an implicit identi�ca-tion of \state" and \continuous state". Compare also the distinnction betweeninternally and externally induced events that was discussed in Chpater 1. Weshall concentrate in particular on cases where the trajectories of the continu-ous state variables are continuous functions of time (no jumps) and the cellsall have equal dimension (no motions along lower-dimensional surfaces).Speci�cally, let the continuous state space X be divided into cells Xi cor-responding to discrete states i 2 I , and suppose that the dynamics in eachmode i is given by the ODE _x(t) = fi(x(t)). To prove for instance stability ofthe origin, a possible strategy is to look for a Lyapunov function V (x) whichin particular should be such that @V@x (x)fi(x) � 0 when x 2 Xi. Finding aLyapunov function for systems with a single mode is already in most cases adi�cult problem; in the multimodal case things can only be expected to beworse in general. Below we shall concentrate attention on situations in whichthe cells are described by linear inequalities and the dynamics in each cell islinear.In the single-mode case, the theory of constructing a Lyapunov function forlinear systems is classical. Given a linear ODE of the form _x(t) = Ax(t) withA 2 Rn�n , one looks for a quadratic Lyapunov function, that is, one of theform V (x) = xTPx where P is a positive de�nite matrix. For such a function,the expression @V@x is a quadratic form that we may write as xT (ATP +PA)x,and so the function V (x) = xTPx quali�es as a Lyapunov function for thesystem _x = Ax if and only if the matrix ATP + PA is negative de�nite.To �nd a Lyapunov function of this form we may solve the linear equationATP + PA = �Q where Q is some given positive de�nite matrix (take forinstance Q = I) and where P is the unknown. A standard result in stabilitytheory asserts that, whenever the matrix A is Hurwitz, this equation has aunique solution which is positive de�nite. So, asymptotic stability of linearsystems can always be proved by a quadratic Lyapunov function, and sucha Lyapunov function can be found by solving a linear system of 12n(n + 1)equations in 12n(n + 1) unknowns. The stability analysis of linear systems istherefore computationally quite feasible, in sharp contrast to the situation forgeneral nonlinear systems.In the multimodal case it is a natural idea to look for piecewise quadraticLyapunov functions when the dynamics in all modes are linear. In general, adynamical system de�ned on Rn is said to be quadratically stable with respectto the origin if there exists a positive de�nite matrix P and a scalar " > 0such that d(xTPx)=dt � �"xTx along trajectories of the system. We shallconcentrate here on quadratic stability with respect to the origin, althoughthere are of course other types of stability, and stability of periodic orbits is in

126 Chapter 5. Analysis of hybrid systemsmany applications perhaps even more interesting than stability of equilibria.Consider now a piecewise linear system; speci�cally, let the dynamics inmode i be given by _x = Aix. The strongest case of stability occurs when allthe modes have a common Lyapunov function. If the Lyapunov function isquadratic, this means that there exists a symmetric positive de�nite matrix Psuch thatATi P + PAi < 0 (5.8)for all i. In this case the stability does not even depend on the switchingscheme. Note that the common Lyapunov function may be found, if it exists,by solving a system of linear matrix inequalities (LMIs).The condition (5.8) requires in particular that all the constituent matri-ces Ai are Hurwitz. One can easily �nd examples, however, in which state-controlled switching of unstable systems leads to an asymptotically stable sys-tem. Consider for instance the two linear systems given by the matricesA1 = 24 0:1 �51 0:1 35 ; A2 = 24 0:1 �15 0:1 35 : (5.9)Figure 5.3 shows trajectories of the systems _x = A1x and _x = A2x startingfrom (1; 0) and (0; 1) respectively. Clearly both systems are unstable. However,if we form the switched system which follows the law _x = A1x on the �rst andthird quadrant and the law _x = A2x on the second and fourth quadrant, thenthe result is asymptotically stable as can be veri�ed in this case by computingthe trajectories of the switched system explicitly. By reversing time in thisexample one gets an example of an unstable system that is obtained fromswitching between two asymptotically stable systems.For a state-controlled switching scheme, the condition (5.8) can be replacedby the weaker conditionfor all i 2 I : xT (ATi P + PAi)x < 0 for all x 2 Xi, x 6= 0: (5.10)which still implies asymptotic stability of the system subject to state-controlledswitching. The above condition is even su�cient for asymptotic stability incase a sliding mode occurs along the boundary of two regions Xi and Xj ,since the conditions xT (ATi P + PAi)x < 0 and xT (ATj P + PAj)x < 0 implyxT (ATP +PA)x < 0 for all convex combinations A of Ai and Aj . A concreteway of using the criterion (5.10) is to look for symmetric matrices Si such thatxTSix � 0 for x 2 Si. The existence of a positive de�nite matrix P such thatfor all iATi P + PAi + Si < 0 (5.11)is then su�cient for asymptotic stability. This is the so-called S-procedure(see e.g. [25]).

5.2. Stability 127

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 5.3: Trajectories of the systems (5.9); initial points marked by stars

128 Chapter 5. Analysis of hybrid systemsTo further weaken the conditions for stability, one may look at Lyapunovfunctions that are piecewise quadratic rather than quadratic. If a region-controlled system of the above type is considered, then a natural idea is tolook for a function of the form xTPix for x 2 Xi, where one should havexTPix = xTPjx when x is in a common boundary of regions Xi and Xj , inorder to make the de�nition consistent and to make the standard stabilityarguments work. As a particular case, consider a system of the form_x = A1x if Cx � 0; _x = A2x if Cx � 0 (5.12)where C is a linear mapping from Rn to R and where it is assumed thatno sliding mode occurs, so that we do not necessarily need to assume thatA1x = A2x when Cx = 0. Stability will be guaranteed if we can �nd symmetricpositive de�nite matrices P1 and P2 such that ATi Pi + PiAi < 0 for i = 1; 2,and xTP1x = xTP2x when Cx = 0. The latter condition is ful�lled when P1and P2 are given by P1 = P , P2 = P + �CTC for some real �. So a su�cientcondition for stability of the system (5.12) is the existence of a matrix P anda scalar � such thatP = P T > 0; AT1 P + PA1 � 0; P + �CTC > 0;AT2 (P + �CTC) + (P + �CTC)A2 � 0: (5.13)Note that (5.13) constitutes a system of linear matrix inequalities. The condi-tions above imply in particular that both matrices and A1 and A2 are stable,so that the suggested method can only used for systems composed of sta-ble subsystems. There is no help from the observation that it is su�cient ifxT (AT1 P + PA1)x � 0 for Cx � 0, and likewise for the other mode, becausethe condition is invariant under change of sign of x. It can be seen from simpleexamples that a system of the form (5.12) may well be asymptotically stableeven if A1 and A2 are not both stable; take for instanceA1 = 24 1 1�1 1 35 ; A2 = 24 �2 1�1 �2 35 (5.14)together with any nonzero functional C.An even further reduction of conservatism can be obtained by dropping therequirement that the Lyapunov function should be continuous; indeed, there isno problem with discontinuities as long as jumps that take place along trajec-tories are downward. Again, this observation is of no help when the consideredLyapunov functions are piecewise quadratic and the cells are halfspaces. Inother cases, however, one may write down the inequalities that should be satis-�ed by a piecewise quadratic but not necessarily continuous Lyapunov functionand attempt to solve the resulting system by LMI techniques.Remark 5.2.3. In the above we have assumed in a number of places thatno sliding modes occur. The presence of motions along lower-dimensional

5.3. Chaotic phenomena 129manifolds may de�nitely a�ect the stability analysis. As shown in the followingsimple example, a sliding mode can be unstable even when it is composed fromtwo stable dynamics.Example 5.2.4. Consider the systems _x = A1x and _x = A2x withA1 = 24 1 2�2 �2 35 ; A2 = 24 1 �22 �2 35 (5.15)and suppose that mode 1 is valid for x2 < 0 and mode 2 for x2 > 0. It iseasily veri�ed that both modes would be asymptotically stable if they wouldbe valid on the whole plane. On the line x2 = 0 there is a sliding mode whichis obtained by averaging the two dynamics, and which results in an unstablemotion _x1 = x1.5.3 Chaotic phenomenaAs is well-known, smooth dynamical systems of dimension three and highermay give rise to complicated behavior. Although there is no universally ac-cepted de�nition of \chaos", the term is usually associated to phenomena suchas sensitive dependence on initial conditions and the presence of attractorscontaining several dense orbits. In the literature, several examples are knownof situations in which a few simple smooth systems (for instance linear sys-tems) give rise to chaotic phenomena when they are combined into a hybridsystem by some switching rule. Here we brie y discuss three of these examples.Example 5.3.1 (A switched arrival system). The following system wasdescribed in [36]. Consider N bu�ers from which work is removed at rates�i (i = 1; : : : ; N). A server delivers work to one of the bu�ers at rate 1. Itis assumed that the total amount of work in the system is constant, so thatPNi=1 �i = 1. As soon as one of the bu�ers becomes empty, the server switchesto that bu�er. The system has N continuous variables xi which denote theamounts of work in each of the bu�ers, and one discrete variable j whichdenotes the bu�er currently being served. The dynamics of the system isgiven by the EFF_x = ej � � jj ji2NfeTi x� = 0; j+ = ig (5.16)where x is the vector of bu�er contents, � is the constant vector of extractionrates, ei denotes the ith unit vector, and N is the set f1; : : : ; Ng. Note thatthe system is not deadlock-free; if two bu�ers empty at the same time, thereis no way to continue. However it is shown in [36] that for almost all initialconditions this never happens.For N = 2, it is easy to verify that the switched arrival system has asimple periodic behavior. However already for N = 3 most trajectories have ahighly irregular behavior, and it can be shown that all periodic trajectories are

130 Chapter 5. Analysis of hybrid systemsunstable. There is some regularity in a statistical sense, however. The mainresult of [36] states that there is one probability distribution which is stationaryin the sense that if the initial condition with one of the bu�ers empty is drawnfrom this distribution, then the distribution of the state at the next eventtime is the same. This distribution gives certain probabilities (depending onthe rates �1, �2 and �3) to each of the three bu�ers being empty, and withinthese cases assigns a uniform distribution over the contents of the other twobu�ers, subject, of course, to the requirement that the sum of the contentsof the bu�ers should be equal to the constant amount of work that is alwayspresent in the system. It is moreover shown that this distribution is statisticallystable, which means that if we start with an initial condition that is distributedaccording to some arbitrary distribution, then the distribution of the state(taken at event times) will converge to the stationary distribution. Finallythe ergodicity property holds. Ergodicity means that for almost all initialconditions \time averages are the same as sample averages", that is to saythat the value of the average over time of any measurable function of the stateat event times can be computed as the expected value of that function withrespect to the stationary distribution. By this result, it can be computed thatfor almost all inital conditions the average interevent time for the trajectorystarting from that initial condition is equal to 14 (�1�2 + �2�3 + �1�3)�1.Example 5.3.2 (A vibrating system with a one-sided spring). In [74]a mechanical system is described that consists of a vibrating beam and a one-sided spring. The spring is placed in the middle of the beam in such a way thatit becomes active or inactive when the midpoint of the beam passes throughits equilibrium position. The system is subject to a periodical excitation froma rotating mass unbalance attached to the beam. The model proposed in [74]has three degrees of freedom. The equations of motion are given byM �q +B _q + (K +H)q = Gf (5.17)if the displacement of the midpoint of the beam is positive, and byM �q +B _q +Kq = Gf (5.18)if the displacement is negative. Here, q is the three-dimensional con�gurationvariable, M is the mass matrix, B is the damping matrix, K is the sti�nessmatrix when the spring is not active, H is the additional sti�ness due to thespring, f is the excitation force, and G is an input vector.The main emphasis in [74] is on control design and no extensive study ofchaotic phenomena is undertaken. Still the analysis shows at least that thereare several unstable periodic orbits. In the cited paper the aim is to add acontroller to the system which will stabilize one of these orbits. Because thecontrol takes place around an orbit that is already periodic by itself (althoughunstable), one may expect that a relatively small control e�ort will be neededfor stabilization once the desired periodic regime has been achieved. This ideaof using only a small control input by making use of the intrinsic nonlinear

5.3. Chaotic phenomena 131properties of the given system is one of the leading themes in the researchliterature on \control of chaos".Example 5.3.3 (The double scroll circuit). A well-known example of asimple electrical network that shows chaotic behavior is Chua's circuit, de-scribed for instance in [38]. The circuit consists of two capacitors, one induc-tor, one linear resistor and one (active) nonlinear resistor. The equations maybe given in the form of a feedback system with a static piecewise linear elementin the feedback loop, as follows:_x1 = �(x1 � u)_x2 = x1 � x2 + x3_x3 = ��x2y = x1 (5.19)and ��������� y � �1; u = m1x� (m0 �m1)�1 � y � 1; u = m0xy � 1; u = m1x+ (m0 �m1): (5.20)Note that the function de�ned in (5.20) is continuous, so that the above systemmay also be looked at as a three-dimensional dynamical system de�ned by apiecewise linear continuous vector �eld. The symbols �, �, m0 and m1 denoteparameters. If one studies the system's behavior in terms of these parametersone �nds a rich variety of motions. For some parameter values, trajectoriesoccur that look like two connected scrolls, and for this reason the system aboveis also known as the \double scroll system". The presence of chaos (in one ofthe possible de�nitions) is shown in [38] for the values m0 = �1=7, m1 = 2=7,� = 7, and � near 8.6.The examples above present just a few instances of chaos in nonsmoothdynamical systems; many more examples can be found in the literature. Wehave chosen examples that exhibit switching between di�erent modes. Theearliest examples of chaos in nonsmooth systems are related to what mightbe called impacting systems in which there are state-generated events but nochanges of mode. The standard example is the one of a ball bouncing on avibrating table, which is discussed in the well known textbook [59]. In contextsof this type it is natural to look for parameter changes that will lead from asituation in which there are no impacts to a situation in which one does have(low-velocity) impacts. The associated bifurcations are called grazing bifurca-tions or border collision bifurcations ; see for instance [32]. Impacting systemscan be seen as limiting cases of situations in which a contact regime exists fora very short time period. In many mechanical applications it is reasonable to

132 Chapter 5. Analysis of hybrid systemsconsider systems that are subject to di�erent regimes for periods of compara-ble length. The one-sided spring mentioned above is an example of this type.Chaotic behavior has also been shown to occur for instance in periodicallyexcited mechanical systems subject to Coulomb friction, and also in this caseone typically sees di�erent regimes that coexist on the same time scale; seefor instance [131]. In an electrical circuit context, the occurrence of chaoticphenomena in systems with diodes, and in particular DC-DC converters, iswell documented; see for instance [42, 47]. In this context one typically haslinear dynamics in each mode. In particular it is noted from these examplesthat linear complementarity systems, which were extensively discussed in theprevious chapter, may exhibit chaotic behavior.5.4 Notes and references for Chapter 5The issue of formal veri�cation has been extensively discussed in computerscience. Various methodologies exist; see for instance [91, 110]. In the par-ticular context of hybrid systems, many papers related to veri�cation can befound in the workshop proceedings [58, 5, 3, 100, 6, 137, 153]; see also [87].A classical text on stability is the book by Hahn [60]; for the special case oflinear systems see also Gantmacher [56]. A switched linear system can some-times be looked at as a smooth system with a nonsmooth feedback. The studyof such systems has a long history; see for instance Popov's book [130]. Inthe control context, the issue of stability is closely related to stabilization.This topic will be discussed in the following chapter. In Subsection 5.2.3 wehave partly followed the paper [83] by Johansson and Rantzer, which givesa nice survey of quadratic stability for hybrid systems. Other references ofinterest in this connection include [126, 127]. The literature on chaos is ofcourse enormous; see for instance [59, 123, 134] for entries into the literature.A speci�c list of references on the analysis of nonsmooth dynamical systemscan be found at a useful website maintained at the University of Cologne(www.mi.uni-koeln.de/mi/Forschung/Kuepper/Forschung1.htm).

Chapter 6Hybrid control designIn this chapter we indicate some directions in hybrid control design. Apartfrom the introduction of a few general concepts, this will be mainly done bythe treatment of some illustrative examples.The area of hybrid control design is very diverse. The challenging topicof controlling general hybrid systems is a wide open problem, and, followingup on our discussion in Chapter 1 on the modelling of hybrid systems, it isnot to be expected that a powerful (and numerically tractable) theory canbe developed for general hybrid systems. Particular classes of hybrid systemsfor which control problems have been addressed include, among others, batchcontrol processes, power converters and motion control systems, as well asextensions of reactive program synthesis to timed automata. Furthermore,in some situations it is feasible to abstract the hybrid system to a discrete-event system (or to an automaton or Petri-net), in which case recourse canbe taken to discrete-event control theory. Since our aim is to expose somegeneral methods, we will not elaborate on these more speci�c areas, but insteadindicate in Section 6.1 some general methods of hybrid control design based ongame theory and viability theory (or, in control theoretic parlance, the theoryof controlled invariance). In particular, we discuss the synthesis of controllersenforcing reachability speci�cations.The main emphasis in this chapter will be on the design of hybrid controllersfor continuous-time systems as ordinarily considered in control theory. From acontrol perspective this subject has classical roots, and we will actually refor-mulate some classical notions into the modern paradigm of hybrid systems. Inparticular, in Section 6.2 we introduce some general terminology on switchingcontrol, and we give relations with the classical notions of pulse width modula-tion and sliding mode control. Furthermore, in continuation of the discussionof quadratic stability in Subsection 5.2.3, we discuss the quadratic stabilizationof multi-modal linear systems. In Section 6.3 we consider the speci�c topic ofstabilizing continuous-time systems by switching control schemes. This, partlyclassical, area has regained new interest by the discovery that there are largeclasses of nonlinear continuous time systems that are controllable and can bestabilized by switching control schemes, but nevertheless cannot be stabilizedby continuous state feedback. Also we will indicate that for controlling certainphysical systems, switching control schemes may be an attractive option which133

134 Chapter 6. Hybrid control designallows for a clear physical interpretation.6.1 Safety and guarantee propertiesLet us consider a hybrid system, either given (see Chapter 1) as a (generalized)hybrid automaton (De�nitions 1.2.3 and 1.2.5), as a hybrid behavior (De�ni-tion 1.2.6), or described by event- ow formulas (Subsection 1.2.6). Considerthe total behavior B of the hybrid system, e.g., in the (generalized) hybridautomaton model, the set of trajectories � of the hybrid system given by func-tions P : �E ! L, x : �E ! X , S : E ! A, w : �E ! W , for time event sets E ,and the corresponding time evolutions �E . A property , P , of the hybrid systemis a mapP : B ! ftrue, falseg:A trajectory � satis�es property P if P(�) = true, and the hybrid systemsatis�es property P if P(�) = true for all trajectories � 2 B.As already indicated in Chapter 5 in the context of veri�cation, althougha general formulation of correctness of a hybrid system should be in termsof language inclusion, certain properties of interest of a hybrid system canbe often expressed more simply as reachability properties of the discrete andcontinuous state variables, so that legal trajectories are those in which certainstates are not reached. Thus, let as above L denote the discrete part of thestate space and X the continuous part of the state space of the hybrid system.Given a set F � (L�X) we de�ne a safety property , denoted by �F , by�F (�) = 8<: true if for all t 2 �E ; (P (t); x(t)) 2 Ffalse otherwiseAlso, we can de�ne a guarantee property , denoted by }F , by}F (�) = 8<: true if there exists t 2 �E such that (P (t); x(t)) 2 Ffalse otherwise(The notations � and } originate from temporal logic, cf. [104].) These twoclasses of properties are dual in the sense that for all subsets F � L�X andall trajectories �}F (�)() :�F c(�)where F c denotes the complement of the set F . Thus, in principle, we mayonly concentrate on safety properties.Remark 6.1.1. Of course, one can still de�ne other properties given a subsetF � L � X . For example, one may require that � will enter the subset F

6.1. Safety and guarantee properties 135at some �nite time instant, and from then on will remain in F . This can beseen as a combination of a guarantee and a safety property. Alternatively,one may de�ne a property by requiring that � enters the set F at in�nitelymany time instants. Also this property can be regarded as a combination ofguarantee and safety (after any �nite time instant we are sure that � will visitF ). From a computational point of view these \combined" properties can oftenbe studied by combining the algorithms for safety and guarantee properties,see (especially for the �nite automaton case) [102].6.1.1 Safety and controlled invarianceFor �nite automata (De�nition 1.2.2) the study of safety properties, as wellas of guarantee properties, is relatively easy. In particular, let us considerautomata given by the following generalization of input-output automata (1.3):l] 2 �(l; i)o = �(l; i) (6.1)where � : L� I ! 2L de�nes the possible state successors l] to a state l giventhe control input i 2 I . This models automata where the transitions are onlypartially controlled, or non-deterministic automata.Let F � L be a given set. Recall that a trajectory � has property �Fif its discrete state trajectory P (t) satis�es P (t) 2 F for all time instants t.This motivates the following invariance de�nition. First, for any initial statel0 at time 0, and a control sequence i0; i1; i2; : : : at the time instants 0; 1; 2; : : : ,let us denote the resulting state trajectories of (6.1) by l0; l1; l2; l3; : : : , wherel1 2 �(l0; i0), l2 2 �(l1; i1), l3 2 �(l2; i2), and so on. A subset V � L is calledcontrolled invariant for (6.1) if for all l0 2 V there exists a control sequencei0; i1; i2; : : : such that any resulting state trajectory l0; l1; l2; l3; : : : of (6.1)remains in V , that is, lj 2 V for j = 0; 1; 2; : : : .De�ne for any subset V � L its controllable predecessor con(V ) bycon(V ) = fl 2 L j 9i 2 I such that �(l; i) � V g: (6.2)It follows that V is controlled invariant if and only ifV � con(V ): (6.3)In order to compute the set of discrete states l 2 F for which there exists acontrol action i0; i1; i2; : : : such that the state will remain in F for all futurediscrete times 0; 1; 2; : : : we thus have to compute the maximal controlledinvariant subset contained in F . This can be done via the following algorithm.V 0 = FV j�1 = F \ con(V j); for j = 0;�1;�2; : : : ; untilV j�1 = V j : (6.4)

136 Chapter 6. Hybrid control designClearly, the sequence of subsets V 0; V �1; V �2; : : : is non-increasing, and so,by �niteness of F , the algorithm converges in at most jF j steps. At each stepof the algorithm, the set V j contains the discrete states for which there existsa sequence of control actions ij ; ij+1; ij+2; : : : ; i�1 which will ensure that thesystem remains in F for at least �j steps. It is easy to see that the subsetobtained in the �nal step of the algorithm, that isV �(F ) := V j�1 = V jis the maximal controlled invariant subset contained in F . Indeed, V �(F ) isclearly a controlled invariant subset of F , since it satis�es by constructionV �(F ) = F \ con(V �(F ))and therefore V �(F ) satis�es (6.3), as well as V �(F ) � F . Furthermore, itcan be shown inductively that any controlled invariant subset V � F satis�esV � V j for j = 0;�1;�2; : : : , and hence V � V �(F ), proving maximality ofV �(F ).Hence for every state l 2 V �(F ) there exists a control action such thatthe resulting trajectory satis�es property �F , and these are precisely all thestates for which this is possible.Note that algorithm (6.4) is constructive also with regard to the requiredcontrol action: in the process of calculating V j�1 we compute for every l 2V j�1 a control value i such �(l; i) � V j . So, at the end of the algorithm wehave computed for every l 2 V �(F ) a sequence of control values which keepsthe trajectories emanating from l in F (in fact, in V �(F )). Furthermore,outside V �(F ) we still have complete freedom over the control action.Using the duality between safety properties and guarantee properties, wecan also immediately derive an algorithm for checking the property }F :S0 = FSj�1 = F [ con(Sj); for j = 0;�1;�2; : : : ; untilSj�1 = Sj : (6.5)Here Sj contains the discrete states from which a visit to F can be enforcedafter at most �j steps. Furthermore, S0; S�1; S�2; : : : is a non-decreasingsequence, and the limiting setS�(F ) := Sj�1 = Sjde�nes the set of locations for which there exists a control action such thatthe resulting trajectory of the �nite automaton satis�es property }F .As noted in Chapter 1 the two most \opposite" examples of hybrid sys-tems are on the one hand �nite automata (De�nition 1.2.2) and continuous-time systems (De�nition 1.2.1). It turns out, roughly speaking, that also forcontinuous-time systems a safety property �F can be checked in a similar

6.1. Safety and guarantee properties 137way as above, provided the subset F of the continuous state space X is asubmanifold of X . Indeed, let us consider for concreteness a continuous-timeinput-state-output system_x = f(x) + g(x)u+ p(x)dy = h(x) (6.6)where, compared with (1.2), we have split the input variables into a set ofcontrol variables u 2 Rm and a set of disturbance variables d 2 Rl . Theinterpretation is similar to (6.1): part of the input variables can be completelycontrolled, while the remaining part is totally uncontrollable. Furthermore,as compared with (1.2) we have assumed for simplicity of exposition thatthe dependence on the input variables u and d is of an a�ne nature via thematrices g(x) and p(x) (see for the general case e.g. [121], Chapter 13).Now, let F be a given submanifold of the state spaceX . Denote the tangentspace of X at a point x 2 X by TxX , and the tangent space of any submanifoldN � X at a point x 2 N by TxN . Furthermore, de�ne G(x) as the subspaceof TxX spanned by the image of the matrix g(x), and similarly P (x) as thesubspace of TxX spanned by the image of the matrix p(x).Consider the algorithmN0 = FN j�1 = fx 2 N j j f(x) + P (x) � TxN j +G(x)gfor j = 0;�1;�2; : : : ; untilN j�1 = N j (6.7)where we assume that the subsets N j�1 � X produced in this way are allsubmanifolds of X (see for some technical conditions ensuring this property, aswell as for computing N j�1 in local coordinates, [121], Chapters 11 and 13). Itis immediately seen that the sequence of submanifoldsN j for j = 0;�1;�2; : : :is non-increasing, that isN0 � N�1 � N�2 � � � �and so by �nite-dimensionality of F the algorithm will converge in at mostdimF steps to a submanifoldN�(F ) := N j�1 = N jsatisfying the propertyf(x) + P (x) � Tx(N�(F )) +G(x); for all x 2 N�(F ) (6.8)Also in this case (see e.g. [121]) it can be shown that N�(F ) is the maximalsubmanifold with this property. For this reason N�(F ) is called the maximalcontrolled invariant submanifold contained in F .

138 Chapter 6. Hybrid control designUnder a constant rank assumption (see again [121], Chapters 11 and 13,for details) it can be shown that (6.8) implies the existence of a (smooth) statefeedback u = �(x) such that the closed-loop system_x = f(x) + g(x)�(x) + p(x)d (6.9)leaves the submanifold N�(F ) invariant, for any disturbance function d(�).Hence, the trajectories of the closed-loop system (6.9) starting inN�(F ) satisfyproperty �F .Algorithm 6.7 can be regarded as an in�nitesimal version of Algorithm 6.4;it replaces the transitions in the �nite automaton by conditions on the velocityvector _x and works on the tangent space of the state space manifold X . Notealso that the �niteness property of F in Algorithm 6.4 has been replaced inAlgorithm 6.7 by the �nite-dimensionality of the submanifold F .Algorithm 6.7 works well provided the set F (as well as all the subsequentsubsets N�1; N�2; N�3; : : : ) is a submanifold, but breaks down for more gen-eral subsets. A fortiori for general hybrid systems it is not clear how to extendthe algorithm.6.1.2 Safety and dynamic game theoryAnother approach to checking safety properties of the form �F is based ondynamic game theory . Indeed, consider a continuous-time system_x = f(x; u; d) (6.10)where, as before in (6.6), u 2 U are the control inputs (corresponding tothe �rst player) and d 2 D are the disturbance inputs (corresponding to thesecond player who is known as the adversary). However, we allow U and Dto be arbitrary sets, not necessarily Rm and Rl , implying that (part of) thecontrol and disturbance inputs may be discrete.Furthermore, contrary to the submanifolds F considered before, we assumethat the safety subset F is given by inequality constraints, that isF = fx 2 X j k(x) � 0g (6.11)where k : X ! R is a di�erentiable function with @k@x (x) 6= 0 on the boundary@K = fx 2 X j k(x) = 0gLet now t � 0, and consider the cost functionJ : X � U �D � R� ! R (6.12)where U and D denote the admissible control, respectively, disturbance func-tions, with the end conditionJ(x; u(�); d(�); t) = k(x(0)): (6.13)

6.1. Safety and guarantee properties 139This function may be interpreted as the cost associated with a trajectory of(6.10) starting at x at time t � 0 according to the input functions u(�) andd(�), and ending at time t = 0 at the �nal state x(0). Furthermore, de�ne thevalue functionJ�(x; t) = maxu2U mind2D J(x; u; d; t): (6.14)Then the setfx 2 X j mint02[t;0] J�(x; t0) � 0g (6.15)contains all the states for which the system can be forced by the control u toremain in F for at least jtj time units, irrespective of the disturbance functiond. The value function J� can be computed, in principle, by standard tech-niques from dynamic game theory (see e.g. [14]). De�ne the (pre-)Hamiltonianof the system (6.10) byH(x; p; u; d) = pT f(x; u; d) (6.16)where p is a vector in Rn called the costate. The optimal Hamiltonian isde�ned byH�(x; p) = maxu2U mind2DH(x; p; u; d) (6.17)If J� is a di�erentiable function of x; t, then J� is a solution of the Hamilton-Jacobi equation�@J�(x; t)@t = H�(x; @J�(x; t)@x ) (6.18a)J�(x; 0) = k(x) (6.18b)It follows that there exists a control action such that the trajectory emanatingfrom a state x has property �F if and only if (compare with (6.15))J�(x; t0) � 0 for all t0 2 (�1; 0]: (6.19)In [99] it is indicated how to simplify this condition by considering a modi�edHamilton-Jacobi equation. Indeed, to prevent states from being relabeled assafe once they have been labeled as unsafe (J�(x; t) being negative for sometime t), one may replace (6.18) by�@J�(x; t)@t = minf0; H�(x; @J�(x; t)@x )g (6.20a)J�(x; 0) = k(x) (6.20b)Assume that (6.20) has a di�erentiable solution J�(x; t) which converges fort! �1 to a function �J . Then the setN�(F ) = fx j �J(x) � 0g

140 Chapter 6. Hybrid control designis the set of initial states of trajectories that, under appropriate control ac-tion, will remain in F for all time. Furthermore, the actual construction ofthe controller that enforces this property needs only to be performed on theboundary of N�(F ).Although in many cases the (numerical) solution of the Hamilton-Jacobiequations (6.18) or (6.20) is a formidable task, it provides a systematic ap-proach to checking safety properties for continuous-time systems, and a start-ing point for checking safety properties for certain classes of hybrid systems,see e.g. [99] for two worked examples.6.2 Switching controlIn some sense, the use of hybrid controllers for continuous-time systems is clas-sical. Indeed, we can look at variable structure control, sliding mode control,relay control, gain scheduling and even fuzzy control as examples of hybrid con-trol schemes. The common characteristic of all these control schemes is theirswitching nature; on the basis of the evolution of the plant (the continuous-time system to-be-controlled) and/or the progress of time the hybrid controllerswitches from one control regime to another.6.2.1 Switching logicOne way of formalizing switching control for a continuous-time input-state-output system is by means of Figure 6.1. Here the supervisor has to decide onthe basis of the input and output signals of the system, and possibly external(reference) signals or the progress of time, which of the, ordinary, continuous-time controllers is applied to the system. The �gure indicates a �nite numberof controllers; however, we could also have a countable set of controllers. Ofcourse there could in fact be a continuum of controllers, but then the resultingcontrol schemes would no longer be referred to as switching control schemes.In many cases the di�erent controllers can all be given the same state space(shared state variables), which leads to the simpler switching control structuregiven in Figure 6.2. In this case the supervisor generates a discrete symbol� corresponding to a controller �c(�) producing the control signal u�. Anexample of a switching control scheme was already encountered in Subsection2.2.8 (the supervisor model), in which case the supervisor is a �nite automatonwhich produces control signals on the basis of the measurement of the outputof the plant and the discrete state of the automaton.The supervisor in Figures 6.1 and 6.2 is sometimes called a switching logic.One of the main problems in the design of a switching logic is that usually itis not desirable to have \chattering", that is, very fast switching. There arebasically two ways to suppress chattering: one is sometimes called hysteresisswitching logic and the other dwell-time switching logic.� (Hysteresis switching logic).

6.2. Switching control 141

m

Controller1Controller2 supervisorControllerm� 1 plantController yu

u1u2um�1umFigure 6.1: Switching control

�splant yu

supervisoru�

��c(�)

Figure 6.2: Switching control with shared controller state

142 Chapter 6. Hybrid control designChoose h > 0. Suppose at some time t0, the supervisor �s has justchanged the value of � to q. Then � is held �xed at the value q unlessand until there exists some t1 > t0 at which �p+h < �q (or (1+h)�p <�q , if a scale-invariant criterion is needed) for some p. If this occursthen � is set equal to p. Here �� denotes some performance criteriumdepending on � and some signal derived from the input and the outputsignals of the continuous-time plant, which is used in the switching logic.Clearly, because of the treshold parameter h no in�nitely fast switchingwill occur. (The idea is similar to the use of a boundary layer around theswitching surface in order to avoid chattering in sliding mode control;see Subsection 6.2.3.)� (Dwell-time switching logic).The basic idea is to have some �xed time � > 0 (called the dwell-time)such that, once a symbol � is chosen by the supervisor it will remainconstant for at least a time equal to � . There are many versions ofdwell-time switching logics.We refer to [115] for further developments and other switching logic schemes.From a general hybrid system point of view a continuous-time system (plant)controlled by switching logic is a hybrid system with continuous state corre-sponding to the continuous state of the plant and the continuous state of thecontroller, together with discrete states residing in the controller. Usually, thetransition from one discrete state to another will not give rise to a jump inthe continuous state of the plant, although it could give rise to a jump in thecontinuous state of the controller (\resetting" the controller). Seen from thispoint of view it is clear that switching control, although being a very gen-eral concept, certainly does not cover all possible hybrid control strategies forcontinuous-time systems. In fact, the following hybrid control example (takenfrom [163]) exhibits jumps in the controlled plant and therefore does not �twithin the switching control scheme.Example 6.2.1 (Juggling robot). Consider a one degree-of-freedom jug-gler, i.e., a system composed of an object (a ball) subject to gravity, whichbounces on a controlled mass (a one degree-of-freedom robot).m1�y1 = �m1gm2�y2 = �m2g + uy1 � y2 � 0_y1(t+k )� _y2(t+k ) = �e[ _y1(t�k )� _y2(t�k )] (6.21)where e 2 [0; 1] is Newton's restitution coe�cient (see also Subsection 2.2.3).Here y1 represents the height of the ball, while y2 represents the vertical co-ordinate of the juggling robot (which is controlled by u). The only way ofin uencing the trajectory of the ball is through the impacts, i.e., at the times

6.2. Switching control 143tk such that y1(tk)�y2(tk) = 0, when the velocity of the ball is reset dependingon its initial velocity and the initial velocity of the robot. Note that in orderto compute the velocity of the ball (and of the robot) just after an impact timeone also needs to invoke the law of conservation of momentum, that is,m1 _y1(t�k ) +m2 _y2(t�k ) = m1 _y1(t+k ) +m2 _y2(t+k ):6.2.2 PWM controlIn some applications one encounters control systems of the following form:_x = f(x; u); x 2 X; u 2 U; (6.22)where X is some continuous space, say Rn , while the input space U is a �nitespace (or, more generally, the product of a continuous and a �nite space). Anappealing class of examples consists of power converters , see e.g. Subsection2.2.13, where the discrete inputs correspond to the switches being closed oropen. Since the input space is �nite such systems can be considered as aspecial case of hybrid systems.In some cases, for instance in power converters, it makes sense to relatethe behavior of these hybrid systems to the behavior of an associated controlsystem with continuous inputs, using the notion of Pulse Width Modulation.Consider for concreteness a control system with U = f0; 1g, that is,_x = f(x; u); u 2 f0; 1g: (6.23)The Pulse Width Modulation (PWM) control scheme for (6.23) can be de-scribed as follows. Consider a so-called duty cycle of �xed small duration �.On every duty cycle the input u is switched exactly one time from 1 to 0. Thefraction of the duty cycle on which the input holds the �xed value 1 is knownas the duty ratio and is denoted by �. The duty ratio may also depend on thestate x (or, more precisely, on the value of the state sampled at the beginningof the duty cycle). On every duty cycle [t; t+�] the input is therefore de�nedby u(�) = 1; for t � � < t+ ��u(�) = 0; for t+ �� � � < t+� (6.24)It follows that the state x at the end of this duty cycle is given byx(t+�) = x(t) + Z t+��t f(x(�); 1)d� + Z t+�t+�� f(x(�); 0)d� (6.25)The ideal averaged model of the PWM controlled system is obtained by lettingthe duty cycle duration � go to zero. In the limit, the above formula (6.25)then yields_x(t) = lim�!0 x(t +�)� x(t)� = �f(x(t); 1) + (1� �)f(x(t); 0) (6.26)

144 Chapter 6. Hybrid control designThe duty ratio � can be thought of as a continuous-valued input, taking itsvalues in the interval [0; 1]. Hence for su�ciently small � the trajectories ofthe continuous-time system (6.26) will be close to trajectories of the hybridsystem (6.23). Note that in general the full behavior of the hybrid system(6.23) will be richer than that of (6.26); not every trajectory of (6.23) can beapproximated by trajectories of (6.26).The Pulse Width Modulation control scheme originates from the controlof switching power converters, where usually it is reasonable to assume thatthe switches can be opened (u = 0) and closed (u = 1) su�ciently fast at anyratio � 2 [0; 1].A similar analysis can be performed for a control system (6.23) with arbi-trary �nite input space U � Rm . Then the PWM-associated continuous-timesystem has continuous inputs � taking values in the convex hull of U .6.2.3 Sliding mode controlA classical example of switching control is variable structure control, as alreadypartly discussed in Subsection 2.2.7 and Chapter 3. Consider a control systemdescribed by equations of the form_x(t) = f(x(t); u(t))where u is the (scalar) control input. Suppose that a switching controlscheme is employed that uses a state feedback law u(t) = �+(x(t)) whenthe scalar variable y(t) de�ned by y(t) = h(x(t)) is positive and a feedbacku(t) = ��(x(t)) whenever y(t) is negative. Writing f+(x) = f(x; �+(x)) andf�(x) = f(x; ��(x)), we obtain a dynamical system that obeys the equa-tion _x(t) = f+(x(t)) on the subset where h(x) is positive, and that fol-lows _x(t) = f�(x(t)) on the subset where h(x) is negative. The surfacefxjh(x) = 0g is called the switching surface. The equations can be takentogether in the form (see (3.3))_x = 12 (1 + v)f+(x) + 12 (1� v)f�(x); v = sgn(h(x)) (6.27)The extension to multivariable inputs u and outputs y is straightforward.Solutions in Filippov's sense (cf. Chapter 3) can be used in the situation inwhich there is a \chattering" behavior around the switching surface h(x) = 0.The resulting solutions will remain on the switching surface for some time.For systems of the form (6.27), where v is the input, the di�erent solutionnotions discussed by Filippov all lead to the same solution, which is most easilydescribed using the concept of the equivalent control veq(x). The equivalentcontrol is de�ned as the function of the state x, with values in the interval[�1; 1], that is such that the vector �eld de�ned by_x(t) = 12 (1 + veq(x))f+(x) + 12 (1� veq(x))f�(x)leaves the switching surface h(x) = 0 invariant (see Chapter 3 for furtherdiscussion).

6.2. Switching control 145In sliding mode control , or brie y, sliding control, the theory of variablestructure systems is actually taken as the starting point for control design.Indeed, some scalar variable s depending on the state x and possibly the timet is considered such that the system has a \desired behavior" whenever theconstraint s = 0 is satis�ed. The set s = 0 is called the sliding surface. Nowa control law u is sought such that the following sliding condition is satis�ed(see e.g. [144])12 ddts2 � ��jsj (6.28)where � is a strictly positive constant. Essentially, (6.28) states that thesquared \distance" to the sliding surface, as measured by s2, decreases alongall system trajectories. Thus it constraints trajectories to point towards thesliding surface. Note that the lefthand side can be also written as s _s; a moregeneral sliding condition can therefore be formulated as designing u such thats _s < 0 for s 6= 0.The methodology of sliding control can be best demonstrated by giving atypical example.Example 6.2.2. Consider the second-order system�q = f(q; _q) + uwhere u is the control input, q is the scalar variable of interest (e.g. the positionof a mechanical system), and the dynamics described by f (possibly non-linearor time-varying) is not exactly known, but estimated by f . The estimationerror on f is assumed to be bounded by some known function F (q; _q):jf � f j � F:In order to have the system track some desired trajectory q(t) = qd(t), wede�ne a sliding surface s = 0 withs = _e+ �ewhere e = q� qd is the tracking error, and the constant � > 0 is such that theerror dynamics s = 0 has desirable properties (e.g. su�ciently fast convergenceof the error to zero). We then have_s = f + u� �qd + � _e:The best approximation u of a continuous control law that would achieve _s = 0is thereforeu = �f + �qd � � _e:Note that u can be interpreted as our best estimate of the equivalent control.In order to satisfy the sliding condition (6.28) we add to u a term that isdiscontinuous across the sliding surface:u = u� k sgn s (6.29)

146 Chapter 6. Hybrid control designwhere sgn is the signum function de�ned in (1.13). By choosing k in (6.29)large enough we can guarantee that the sliding condition (6.28) is satis�ed.Indeed, we may take k = F + �. Note that the control discontinuity k acrossthe sliding surface s = 0 increases with the extent of parametric uncertaintyF . The occurrence of a sliding mode may not be desirable from an engineeringpoint of view; depending on the actual implementation of the switching mech-anism, a quick succession of switches may occur which may lead to increasedwear and to high-frequency vibrations in the system. Hence for the actualimplementation of sliding mode control it is usually needed, in order to avoidvery fast switching, to embed the sliding surface in a thin boundary layer, suchthat switching will only occur outside this boundary layer (hysteresis switchinglogic). Furthermore, the discontinuity sgn s in the control law can be furthersmoothened, say, by replacing sgn by a steep sigmoid function. Of course, suchmodi�cations may deteriorate the performance of the closed-loop system.One of the main advantages of sliding mode control, in addition to its con-ceptual simplicity, is its robustness with respect to uncertainty in the systemdata. A possible disadvantage is the excitation of unmodeled high-frequencymodes.6.2.4 Quadratic stabilization by switching controlIn continuation of our discussion of quadratic stability in Chapter 5, let us nowconsider the problem of quadratic stabilization of a multi-modal linear system_x = Aix; i 2 I; x 2 Rn (6.30)where I is some �nite index set. The problem is to �nd a switching rule suchthat the controlled system has a single quadratic Lyapunov function xTPx. Itis not di�cult to show that this problem can be solved if there exists a convexcombination of the matrices Ai, i 2 I , which is Hurwitz (compare with thecorresponding discussion in Chapter 5, Subsection 5.2.3). To verify this claim,assume that the matrixA := X�iAi (�i � 0; P�i = 1) (6.31)is Hurwitz. Take a positive de�nite matrix Q, and let P be the solution ofthe Lyapunov equation ATP +PA = �Q; because A is Hurwitz, P is positivede�nite. Let now x be an arbitrary nonzero vector. From xT (ATP+PA)x < 0it follows thatXi �i[xT (ATi P + PAi)x] < 0: (6.32)Because all the �i are nonnegative, it follows that at least one of the num-bers xT (ATi P + PAi)x must be negative, and in fact we obtain the stronger

6.3. Hybrid feedback stabilization 147statement[i2Ifx j xT (ATi P + PAi)x � � 1N xTQxg = Rn (6.33)where N denotes the number of modes.It is clear how a switching rule for (6.30) may be chosen such that asymp-totic stability is achieved; for instance the rulei(x) := arg minxT (ATi P + PAi)x (6.34)is an obvious choice.The minimum rule (6.34) indeed leads to an asymptotically stable system,which is well-de�ned if one extends the number of discrete states so as toinclude possible sliding modes. To avoid sliding modes, the minimum rulemay be adapted so that the regions in which di�erent modes are active willoverlap rather than just touch. For instance, a modi�ed switching rule (basedon hysteresis switching logic) may be chosen which is such that the systemwill stay in mode i as long as the continuous state x satis�esxT (ATi P + PAi)x � � 12N xTQx: (6.35)When the bound in (6.35) is reached, a switch will take place to a new modej that may for instance be determined by the minimum rule. At the timeat which the new mode j is entered, the number xT (ATj P + PAj)x must beless than or equal to � 1N xTQx. Suppose that the switch to mode j occurs atcontinuous state x0. The time until the next mode switch is given by�j(x0) = minft � 0 j xT0 etATj [ATj P + PAj + 12NQ]etAjx0 � 0g (6.36)(taken to be in�nity if the set over which the minimum is taken is empty).Note that �j(x0) is positive, its dependence on x0 is lower semi-continuous,and satis�es �j(�x0) = �j(x0) for all nonzero real �. Therefore there is apositive lower bound to the time the system will stay in mode j, namelyTj := minf�j(x0) j kx0k = 1; xT0 (ATi P + PAi)x0 = � 12N xT0 Qx0g:(6.37)So under this switching strategy the system is asymptotically stable and therewill be no chattering.6.3 Hybrid feedback stabilizationIn this section we treat a few examples of (simple) continuous-time systemsthat can be stabilized by hybrid feedback, so as to illustrate the potential ofhybrid feedback schemes.

148 Chapter 6. Hybrid control design6.3.1 Energy decrease by hybrid feedbackFirst we consider the well-known harmonic oscillator, with equations given as(after a normalization of the constants)_q = v_v = �q + uy = q (6.38)It is well-known that the harmonic oscillator cannot be asymptotically stabi-lized using static output feedback. However, since the system is controllableand observable, it can be asymptotically stabilized using dynamic output feed-back. (One option is to build an observer for the system, and to substitutethe estimate of the state produced by this observer into the asymptoticallystabilizing state feedback law.) As an illustration of the novel aspects createdby hybrid control strategies, we will show (based on [9]) that the harmonicoscillator can be also asymptotically stabilized by a hybrid static output feed-back.Example 6.3.1. Consider the following controller automaton for the har-monic oscillator. It consists of two locations, denoted + and �. In eachlocation we de�ne a continuous static output feedback u+(q), respectivelyu�(q) byu+(q) = 8<: 0 for q � 0�3q for q < 0 (6.39)and u�(q) = 8<: �3q for q � 00 for q < 0 (6.40)Furthermore, we de�ne a dwell-time T+, respectively T� for each location,which speci�es how long the controller will remain in a location before a tran-sition is made, either to the other or to the same location. The value of q atthe transition time is denoted by qtr. The controller automaton is depicted inFigure 6.3.To verify that such a hybrid feedback asymptotically stabilizes (6.38), con-sider the energy function H(q; p) = 12 (q2 + v2). In the region where u = 0there is no change in the energy. When u(q) = �3q, the time-derivative of theenergy along the solution isddt (12(q2(t) + v2(t))) = �3q(t)v(t)so that the energy decreases if q � v > 0. As shown in Figure 6.3, after atransition the input u = �3q is in operation only when the sign of q(t) � v(t)

6.3. Hybrid feedback stabilization 149T� = 34u�(q)� qtr < 0u+(q)T+ = 34+qtr � 0 qtr < 0

qtr � 0Figure 6.3: Hybrid feedback for the harmonic oscillatorchanges from negative to positive, and then for at most 34 time units, in whichq(t) � v(t) remains positive. (Note that with u = �3q the solutions q(t) andv(t) of the closed-loop system are shifted sinusoids with frequency 2; hencethe product q(t) � v(t) is a shifted sinusoid with frequency 4. However, since�4 > 34 , q(t) � v(t) will not change sign during the time the input u = �3q is inoperation.)An alternative asymptotically stabilizing hybrid feedback, this time hav-ing three locations is provided by the following controller automaton depictedin Figure 6.4, with � > 0 small. In order to verify that this hybrid feed-

qtr � 0qtr < 0

qtr < 0u = �3q

T� = �

d qtr � 0+qtr � 0 qtr < 0�

Td = �4 � 2�u = 0T+ = �u = 0Figure 6.4: An alternative hybrid feedbackback also stabilizes the harmonic oscillator, consider again the total energyH(q; v) = 12 (q2 + v2). At the locations + and � there is no change in energy.These two locations serve to detect a shift of q from positive to negative inthe case of location +, and from negative to positive in the case of location

150 Chapter 6. Hybrid control design�. In both cases the change in sign will occur within a bounded number oftransition periods. Then the location is shifted to d, which serves as a dissi-pation location. Furthermore, since during both + and � the trajectory turnsclockwise, when d is switched on, the initial condition starts at either the �rstor the third orthant (i.e. q � v > 0), and since Td = �4 � 2�, the solutionstays in that orthant as long as d is on. Furthermore, when the location d isreached after a transition, there is a loss of order of 34 of the energy (up to �)before one of the locations + or � is switched on, depending on whether q � 0or q < 0, and the search for the change of the sign starts again. All in all,once + is on with q � 0, or � is on with q < 0, the hybrid trajectory passesthrough d in�nitely often, with bounded gaps between occurrences, and eachtime there is a proportional loss of energy of order close to 34 . The conclusionis that H(q; v) tends to zero, uniformly for q(0); v(0) in a compact set. Thisproves the asymptotic stabilization, provided the hybrid initial condition sat-is�es q � 0 in location +, or q < 0 in location �. It is straightforward to checkthat starting from an arbitrary initial condition, the hybrid trajectory reachesone of these situations within an interval of length �2 , and thus asymptoticstability is shown.The main di�erence between both hybrid feedback stabilization schemes isthat in the �rst case we had only two locations with a nonlinear control, whilein the second case we only employ linear feedback at the price of having threelocations. Furthermore, one could compare the convergence and robustnessproperties of both schemes, as well as compare the two schemes with a dynamicasymptotically stabilizing output feedback.As a preliminary conclusion of this subsection we note that the above hy-brid feedback stabilization strategies for the harmonic oscillator are based onenergy considerations. Indeed, the hybrid feedback schemes were constructedin such a manner that the energy is always non-increasing along the hybridtrajectories. The di�culty of the actual proof of asymptotic stability lies inshowing that the energy is indeed in all cases non-increasing, and that more-over eventually all energy is dissipated. (The idea is somewhat similar toLaSalle's version of Lyapunov stability theory: the Lyapunov function shouldbe always non-increasing along trajectories, while there should be no non-trivial invariant sets where the Lyapunov function remains constant.) Thetools for showing this, however, are rather ad hoc. Furthermore, there is notyet a systematic procedure to construct hybrid feedback stabilization strate-gies. The main problem seems to be that in the hybrid case we do not havean explicit expression of the energy decrease analogous to the time-derivativeddtH = @H@x (x)f(x; u) of the energyH for a continuous-time system _x = f(x; u).Clearly, the idea of a non-increasing energy can be extended from physicalenergy functions H to general Lyapunov functions V (with the ubiquitousdi�culty of �nding feasible Lyapunov functions). The setting seems somewhatsimilar to the use of control Lyapunov functions in the feedback stabilizationof nonlinear control systems, see e.g. [143].

6.3. Hybrid feedback stabilization 1516.3.2 Stabilization of nonholonomic systemsWhile the above example of stabilizing a harmonic oscillator by hybrid staticoutput feedback may not seem very convincing, since the goal could be alsoreached by other means, we now turn to a di�erent class of systems wherethe usefulness of hybrid feedback stabilization schemes is immediately clear.Indeed, let us consider the example of the nonholonomic integrator_x = u_y = v_z = xv � yu (6.41)where u and v denote the controls. To indicate the physical signi�cance ofthis system it can be shown that any kinematic completely non-holonomicmechanical system with three states and two inputs can be converted into thisform (see e.g. [118]).The nonholonomic integrator is the prototype of a nonlinear system, whichis controllable, but nevertheless cannot be asymptotically stabilized using con-tinuous state feedback (static or dynamic). The reason is that the nonholo-nomic integrator violates what is called Brockett's necessary condition [29],which is formulated in the following theorem.Theorem 6.3.2. Consider the control system_x = f(x; u); x 2 Rn ; u 2 Rm ; f(0; 0) = 0; (6.42)where f is a C1 function. If (6.42) is asymptotically stabilizable (about 0) usinga continuous feedback law u = �(x), then the image of every open neighborhoodof (0; 0) under f contains an open neighborhood of 0.Remark 6.3.3. Since we allow for continuous feedback laws (instead of C1or locally Lipschitz feedback laws) some care should be taken in connectionwith the existence and uniqueness of solutions of the closed-loop system.It is readily seen that the nonholonomic integrator does not satisfy thecondition mentioned in Theorem 6.3.2 (Brockett's necessary condition), de-spite the fact that it is controllable (as can be rather easily seen). Indeed,(0; 0; �) 62 Im (f) for any � 6= 0, and so the nonholonomic integrator cannot bestabilized by a time-invariant continuous feedback.In fact, the nonholonomic integrator is an example of a whole class ofsystems, sharing the same property. For example, actuated mechanical systemssubject to nonholonomic kinematic constraints (like rolling without slipping)do not satisfy Brockett's necessary condition, but are often controllable.For the nonholonomic integrator we consider the following sliding modecontrol law (taken from [21]):u = �x+ y sgn zv = �y � x sgn z (6.43)

152 Chapter 6. Hybrid control designConsider a Lyapunov function for the (x; y)-subspace:V (x; y) = 12 (x2 + y2)The time-derivative of V along the trajectories of the closed-loop system (6.41{6.43) is negative:_V = �x2 + xy sgn z � y2 � xy sgn z = �(x2 + y2) = �2V: (6.44)It already follows that the variables x; y converge to zero. Now let us considerthe variable z. Using equations (6.41{6.43) we obtain_z = xv � yu = �(x2 + y2) sgn z = �2V sgn z (6.45)Since V does not depend on z and is a positive function of time, the absolutevalue of the variable z will thus decrease and will be able to reach zero in �nitetime provided the inequality2 Z 10 V (�)d� > jz(0)j (6.46)holds. If this inequality is an equality, then z(t) converges to the origin inin�nite time. Otherwise, it converges to some constant non-zero value thathas the same sign as z(0). After reaching zero z(t) will remain there, sinceaccording to (6.45), all trajectories are directed towards the surface z = 0 (thesliding surface), while x and y always converge to the origin while within thissurface.From (6.44) it follows thatV (t) = V (0)e�2t = 12(x2(0) + y2(0))e�2t:Substituting this expression in (6.46) and integrating, we �nd that the condi-tion for the system to be asymptotically stable is that12 (x2(0) + y2(0)) � jz(0)j:The inequality12 (x2 + y2) < jzj (6.47)de�nes a parabolic region in the state space. The above derivation is summa-rized in the following theorem.Theorem 6.3.4. If the initial conditions for the system (6.41) do not belongto the region de�ned by (6.47), then the control (6.43) asymptotically stabilizesthe system.

6.3. Hybrid feedback stabilization 153If the initial data are inside the parabolic region de�ned by (6.47), we canuse any control law that �rst steers it outside. In fact, any nonzero constantcontrol can be applied. Namely, if u = u0 = const, v = v0 = const, thenx(t) = u0t+ x0y(t) = v0t+ y0z(t) = x(t)v0 � y(t)u0= t(x0v0 � y0u0) + z0With such x, y and z the left hand side of (6.47) is quadratic with respect totime t, while the right hand side is linear. Hence, the state will always leave theparabolic region de�ned by (6.47), and then we switch to the control given by(6.43). Note that the resulting control strategy is inherently hybrid in nature:�rst we apply the constant control u = u0, v = v0 if the state belongs to theparabolic region de�ned by (6.47), and if the system is outside this region thenwe switch to the sliding control (6.43).A possible drawback of the above hybrid feedback scheme is caused by theapplication of sliding control: in principle we will get chattering around thesliding surface z = 0. Although this can be remedied with the usual tools insliding control (e.g. putting a boundary layer around the sliding surface, orapproximating the sgn function by a steep sigmoid function), it motivates thesearch for an alternative hybrid feedback scheme that completely avoids thechattering behavior.Remark 6.3.5. An alternative sliding mode feedback law can be formulatedas follows:u = �x+ yx2 + y2 sgn zv = �y � xx2 + y2 sgn z: (6.48)This feedback will result in the same equation (6.44) for the time-derivative ofV , while the equation (6.45) for the time-derivative of z is improved to_z = � sgn z: (6.49)Of course, the price that needs to be paid for this more favorable behavior isthat the control (6.48) is unbounded for x2 + y2 tending to zero.In connection to the above hybrid feedback schemes (6.43) and (6.48) weconsider the following alternative scheme, taken from [77, 78]. Let q = (x; y; z),and de�ne W (q) = (w1(q); w2(q)) 2 R2 byw1(q) = z2w2(q) = x2 + y2

154 Chapter 6. Hybrid control designWe now de�ne four regions in R2 , and construct the following hybrid feed-back with four locations.1. Pick four continuous functions �k : [0;1)! R; k 2 I = f1; 2; 3; 4g withthe following properties.(a) �k(0) = 0 for each k 2 f1; 2; 3; 4g.(b) for each k 2 f1; 2; 3; 4g and for all w > 00 < �1(w) < �2(w) < �3(w) < �4(w):(c) �1 and �2 are bounded.(d) �1 is such that if w ! 0 exponentially fast then w�1(w) ! 0 expo-nentially fast.(e) �4 is di�erentiable on some non-empty, half-open interval (0; c] andd�4dw (w) < �4(w)w ; w 2 (0; c]:Moreover, if w ! 0 exponentially fast, then �4(w) ! 0 exponen-tially fast.2. Partition the closed positive quadrant � R2 into four overlappingregionsR1 = f(w1; w2) 2 : 0 � w2 < �2(w1)gR2 = f(w1; w2) 2 : �1(w1) < w2 < �4(w1)gR3 = f(w1; w2) 2 : w2 > �3(w1)gR4 = f(0; 0)g:3. De�ne the control law(u(t); v(t))T = g�(t)(q(t))where �(t) is a piecewise constant switching signal taking values in I =f1; 2; 3; 4g andg1(q) = 241135g2(q) = 2664x+ yzx2 + y2y � xzx2 + y23775

6.3. Hybrid feedback stabilization 155g3(q) = 2664�x+ yzx2 + y2�y � xzx2 + y23775g4(q) = 240035 :The signal �(t) is de�ned to be continuous from the right and is deter-mined recursively by�(t) = �(q(t); ��(t)); ��(t0) = �0 2 Iwhere ��(t) denotes the limit of �(t) from below and � : R3 � I ! I isthe transition function�(q; �) = 8<: � if W (q) 2 R�maxfk 2 I :W (q) 2 Rkg otherwiseA typical example is obtained by taking �1(w1) = (1 � e�pw1), �2 = 2�1,�3 = 3�1, and �4 = 4�1. The resulting regions R1; R2 and R3 are shown inFigure 6.5.�������������������������������������������������������������������������������������������

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Figure 6.5: A typical partition of Although the closed-loop system is not globally Lipschitz, global existenceof solutions can be easily established. Indeed, simple calculations show that_w1 � 2w1 + w2; _w2 � 2w2 + 2:Since the bounds for the right-hand sides of the above equations are globallyLipschitz with respect to w1 and w2, these variables and their derivatives arebounded on any �nite interval, from which the global existence of solutionscan be established (see [78] for details). Note that if we start in region R1,then we apply the constant control u = 1 and v = 1 as in the previous hybrid

156 Chapter 6. Hybrid control designsliding mode design. For the same reason as given above, this will lead us tothe region R2. Furthermore, if the switching signal is equal to 2 or 3 we obtain(compare with (6.49))_w1 = �2w1and so w1, and hence z, will converge to zero. Detailed analysis (cf. [78])indeed shows this to happen, and moreover shows that also x and y convergeto zero. The convergence of all variables is exponential.Let us try to summarize what we have learned from this example of hybridfeedback stabilization of a nonholonomic system. As a general remark, thereseems to be at the moment no general design methodology for constructinghybrid feedback stabilization schemes. The �rst scheme for asymptotic stabi-lization of the nonholonomic integrator that we discussed is based on slidingcontrol, while also the second scheme takes much inspiration from such slidingcontrol schemes. However it is not easy to extract a systematic design method-ology from this example. Furthermore, the actual proofs that the proposedhybrid feedback schemes do work are typically complicated and rather ad hoc.6.3.3 Set-point regulation of mechanical systems by en-ergy injectionIn analogy with Subsection 6.3.1 let us continue with hybrid feedback strategiesthat are based on energy considerations. Instead of decreasing the energy ofthe system in bringing the system to a desired rest con�guration, there arealso cases where actually we like to \inject" energy into the system in ane�cient manner. A typical example of this is to swing up a pendulum fromits hanging position to the upright position, cf. [10]. The equations of motionof a pendulum are given byJp�� �mgl sin � +mul cos � = 0: (6.50)Here m denotes the mass of the pendulum, concentrated at the end, l denotesthe length, and Jp is the moment of inertia with respect to the pivot point.The angle between the vertical and the pendulum is denoted by �, where � ispositive in the clockwise direction. The acceleration due to gravity is g and thehorizontal acceleration of the pivot point is u. Note that the linearized systemsare controllable, except when � = �2 or ��2 , i.e. except when the pendulum ishorizontal. One way to swing the pendulum to its upright position is to injectinto the system an amount of energy such that the total energy corresponds tothe potential energy of the upright position. Once the pendulum is su�cientlyclose to the upright position one then switches to a (say, linear) controller inorder to keep the system near this upright equilibrium|so this is already aswitching control strategy! The energy of the system is given byE = 12Jp _�2 +mgl cos � (6.51)

6.3. Hybrid feedback stabilization 157Computing the derivative of E with respect to time we �nddEdt = Jp _��� �mgl _� sin � = mul _� cos � (6.52)It follows from (6.52) that it is easy to control the energy. Only for _� cos � = 0,that is, for _� = 0, � = �2 , or � = ��2 , controllability of the energy increaseis lost. Physically these singular states correspond to the case when the pen-dulum reverses its velocity or when it is horizontal. Control action is moste�ective when the angle � is close to 0 or � and the velocity is large. To in-crease energy the horizontal acceleration of the pivot u should be positive whenthe quantity _� cos � is negative. To change the energy as quickly as possiblethe magnitude of the control signal should be as large as possible.Let the desired energy be E0. The following sliding control law is a simplestrategy for achieving the desired energy:u = satc(k(E �E0)) sgn( _� cos �) (6.53)where k is a design parameter. In this expression the function satc saturatesthe control action at the level c. The strategy is essentially a bang-bangstrategy for large errors and a proportional control for small errors. For largevalues of k the strategy (6.53) is arbitrarily close to the strategy that gives themaximum increase of energy.Let us now discuss strategies for bringing the pendulum from rest in down-ward position to a stationary position in the upright equilibrium. The potentialenergy of the pendulum is normalized to be zero at the upright position andis thus equal to �2mgl at the downward position. One way to swing up thependulum is therefore to control it in such a way that its energy increases from�2mgl to 0. A very simple strategy to achieve this is as follows. Acceleratethe pendulum with maximum acceleration in an arbitrary direction and re-verse the acceleration when the velocity becomes zero. It can be seen that thisstrategy is optimal as long as the pendulum does not reach the horizontal po-sition, because it follows from (6.52) that the acceleration should be reversedwhen the pendulum reaches the horizontal position.The behavior of the swing depends critically on the maximum accelerationc of the pivot. If c is large enough the pendulum can be brought up in oneswing, but otherwise multiple swings are required.In fact it can be seen that if c � 2g then one swing is su�cient by �rstusing the maximum acceleration until the desired energy is obtained, and thensetting the acceleration to zero. (Note that with c = 2g this comes down tomaximum acceleration until the pendulum is horizontal, and then switchingo� the acceleration.) However, using more than two switches as above, itcan be seen that it is possible to bring the pendulum in upright position if43g � c < 2g. In fact, if c = 43g then we maximally accelerate till the pendulumis horizontal, and then we maximally reverse the acceleration until the desiredenergy is reached, at which moment we set the acceleration equal to zero.It is interesting to compare these energy-based switching strategies withminimum time strategies. Indeed, from Pontryagin's Maximum principle it

158 Chapter 6. Hybrid control designfollows that minimum time strategies for juj bounded by c are of bang-bangtype as well. It can be shown that these strategies also have a nice interpre-tation in terms of energy. They will inject energy into the pendulum at amaximum rate and then remove energy at maximum rate in such a way thatthe energy corresponds to the equilibrium energy at the moment when theupright position is reached. In fact, for small values of c the minimum-timestrategies produce control signals that at �rst are identical to those producedby the energy-based strategies. The �nal part of the control signals, however,is di�erent, because the energy-based control strategies will set the accelera-tion equal to zero when the desired energy has been obtained, while in theminimum-time strategies there is an \overshoot" in the energy.In principle, the same reasoning can be used for set-point regulation ofother mechanical systems.6.4 Notes and References for Chapter 6Section 6.1 is mainly based on [99] and [102]. The theory of controlled invari-ance for linear systems can be found in [160], [15], and for the nonlinear casee.g. in [121]. In a more general context the property of controlled invariancehas been studied as \viability theory", cf. [11]. Conceptually very much re-lated work in the area of discrete-event systems is the work of Wonham andco-workers, see e.g. [132], [149]. For optimal control of hybrid systems, in par-ticular the extension of the Maximum Principle to the hybrid case, we referto [147]. Subsection 6.2.1 is largely based on [115], while part of Subsection6.2.3 is based on the exposition of sliding mode control in [144]. There is asubstantial body of literature on the topic of quadratic stabilization that wetouched upon in Subsection 6.2.4. A few references are [103], [128], and [154]where the switching rule according to (6.35) is suggested. Subsection 6.3.1 istaken from [9]. The �rst part of Subsection 6.3.2 is based on [21], see also [22].The second part of Subsection 6.3.2 is based on [78], see also [77]. Finally,Subsection 6.3.3 is based on [10]. Of course, there is much additional workon the control of hybrid systems that we did not discuss here. For furtherinformation the reader may refer to the papers in the recent special issues [7]and [116]. A survey of hybrid systems in process control is available in [94].

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Indexabstraction, 47, 112, 133accumulation point, 10, 17active index set, 93, 96activities, 7AD converter, 46admissible velocity, 82adversary, 118a�ne, 94autonomous jump, 9autonomous switching, 9averaging, 122bicausal, 102border collision bifurcation, 131bouncing ball, 37Brockett's necessary condition, 151Carath�eodory equations, 58Carath�eodory solution, 45cell, 125chaos, 129control of, 131chattering, 58chattering approximation, 61communication variable, 18commutator, 122complementarity problemdynamic, 93, 94linear, 72, 94, 103, 109nonlinear, 93rational, 98, 103complementarity system, 71Hamiltonian, 75, 77linear, 77, 96, 103bimodal, 102nonlinear, 95passive, 74

complementary variables, 71compositionality, 14computer-aided veri�cation, 111concurrent processes, 34consistent state, 97consistent subspace, 97constrained pendulum, 54continuous state variable, 3continuous-time systems, 3control Lyapunov functions, 150controllable predecessor, 135controlled invariant, 135controlled jump, 9controlled switching, 9correct, 111, 113cost function, 138Coulomb friction, 49, 64, 80DA converter, 46DC-DC converter, 132DCP, see complementarity prob-lem, dynamicdeadlock, 10decoupling matrix, 94deterministic input-output au-tomata, 5di�erential inclusion, 67di�erential-algebraic equations, 4,55diode, 52, 71, 73discrete states, 6double scroll system, 131duration, 12duty cycle, 143duty ratio, 124, 143dwell-time switching logic, 142dynamic game theory, 138171

172 Indexearly exercise, 85EFF, see event- ow formulaembedded systems, 2energy, 75, 106, 120equilibrium, 81, 119equivalent control method, 59, 60,63, 64ergodicity, 130event, 4, 7, 9externally induced, 9, 27internally induced, 9, 27multiple, 10event clause, 19event time, 9, 15event- ow formula, 19event-tracking method, 26execution, 8exercise price, 84existence of solutionsglobal, 90local, 90external variable, 3�nite automaton, 4 ow clause, 19formal veri�cation, 111free boundary, 84Fuller phenomenon, 90gain scheduling, 57, 79generalized hybrid automaton, 12geometric inequality constraints,74grazing bifurcation, 131guarantee property, 134Guard, 7Hamilton-Jacobi equation, 139Hamiltonian, 65, 75, 76, 104high contact, 86Hurwitz, 123hybrid automaton, 6hybrid behavior, 17hybrid system, 1hyperstability, 121hysteresis, 35

hysteresis switching logic, 140impacting systems, 131implementation relation, 112implicit Euler method, 88impulsive forces, 105impulsive-smooth distributions,100index, 79initial solution, 101initially nonnegative, 101input symbol, 5input variable, 4input-output automaton, 135input-state-output system, 4interconnection, 14interleaving parallel composition,14invariant, 119inward normal, 82juggling robot, 142Jump, 7jump linear system, 124jump rule, 106jump space, 98Kilmister's principle, 24labels, 5Lagrangian, 76Laplace transform, 100LCP, see complementarity prob-lem, linearleading Markov parameter, 110leading row coe�cient matrix, 102lexicographically nonnegative, 92Lie derivative, 94linear matrix inequality, 126livelock, 10location, 6location invariant, 7Lyapunov function, 118, 125multiple, 121Markov chain, 124Markov parameter, 102

Index 173leading, 102mathematical economics, 81max algebra, 90max-plus system, 89maximal controlled invariant sub-manifold, 137maximal controlled invariant sub-set, 136method of lines, 87minor, 72mode, 53, 96mode selection, 27, 91, 98model checking, 112symbolic, 112multi-body systems, 50multiple collisions, 42multiple time event, 15multiplicity, 15, 103Newton's cradle, 42NLCP, see complementarity prob-lem, nonlinearnon-deterministic automaton, 135non-Zeno, 23nonholonomic integrator, 151object-oriented modeling, 3optimal control, 75, 90optimization, 65option pricing, 84output symbol, 5output variable, 4P-matrix, 72, 102parallel composition, 21, 124passivity, 73, 95perp (?), 71, 99persistent-mode convention, 24, 36piecewise linear system, 79, 109,131Poincar�e mapping, 118, 121power converter, 52, 143principal minor, 72, 110projected dynamical system, 81property, 134Pulse Width Modulation, 143

put optionAmerican, 85European, 84quadratic stabilization, 146quantization, 47railroad crossing example, 51RCP, see complementarity prob-lem, rationalreachability, 113, 115, 117, 134reinitialization, 103relative degree, 94relay, 50, 64, 65, 78, 109restitution coe�cient, 37return map, 121Riemannian manifold, 83right uniqueness, 90right-Zeno, 23, 102row proper, 102run, 8safety property, 134semantics, 5semi-explicit, 72sgn, see signumsigmoid, 61signum, 28, 60, 146simplest convex de�nition, 59, 60,63, 64simulation, 24, 118slack, 110sliding mode, 57, 66, 126, 128sliding mode control, 145sliding motion, 59smooth approximation, 61smooth pasting, 86smoothing method, 25stable, 119asymptotically, 119Lyapunov, 119quadratically, 125statistically, 130state space, 3, 4stepwise re�nement, 112storage function, 95

174 Indexsupervisor, 46, 140switching curve, 66switching logic, 140switching surface, 57synchronous parallel composition,14temporal logic, 112, 117theorem proving, 112thermostat, 38time event, 15time evolution, 15, 16time of maturity, 84timed automata, 117timed Petri net, 89timestepping method, 28totally invertible, 99trajectory, 8, 13, 16transition rule, 4transitions, 4two-carts example, 47, 106unilateral constraints, 74uniqueness of solutions, 90global, 90local, 90value function, 139Van der Pol oscillator, 55variable-structure system, 44, 57,77variational inequality, 81veri�cation, 117viability theory, 158Vidale-Wolfe model, 65volatility, 85water-level monitor, 40well-posed, 11, 25, 67, 95, 96, 102,103Wiener process, 85Zeno behavior, 10